The seven category tabs, from Basic Sets to Relations, switch the row of formula buttons below them; the selected identity stays shaded while you browse. Learn more about the category tabs
The diagram shades some of its four regions: outside both circles, only in A, only in B, and the lens A∩B; hover a region to see its name. Relations identities redraw the circles nested, apart or on top of each other. Learn more about reading the shaded diagram
← Previous and Next → step through all 19 identities in tab order and wrap around; the counter shows the position, such as 1 / 19. Learn more about Previous and Next
The Explanation panel names the identity; its Overview tab gives the definition and, for some identities, an example, and its Learn More tab links to that identity's section on this page. Learn more about getting started
Open the explorer and a two-circle Venn diagram appears with the first identity pre-selected. The blue shaded region marks the elements that satisfy the current identity; the unshaded regions are excluded. The symbol of the current identity appears in the badge above the diagram, and the explanation panel beside it describes what the highlighted region means.
The interface has three main controls. The category tabs at the top group identities by type. The formula buttons below the tabs show the identities within the active category. The Jump to dropdown on the right lists every identity across all categories in one place.
At the bottom of the diagram column, Previous and Next cycle through all 19 identities in order, with a counter showing your current position. The theme panel underneath lets you customize the shading color and opacity.
No setup is required — pick any tab and any button to see the corresponding region light up immediately.
Navigating Category Tabs
DemoCategory tabs and formula buttons
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The category tabs organize all 19 identities into seven groups based on their structural role:
• Basic Sets — the setsA and B themselves, the universal set U, and the empty set∅ • Complements — A′ and B′, also written Ac and Bc in the set operations lessons; the prime is this tool's house style and the two spellings mean the same thing • Intersection & Union — A∩B and A∪B • Differences — A∖B, B∖A, and the symmetric difference A△B • Compound — combined expressions like A∪B′ and A′∪B • De Morgan's Laws — (A∪B)′ and (A∩B)′ • Relations — subset, disjoint, and equal sets, each drawn with a special circle layout
Click a tab to switch the row of formula buttons below. The currently selected identity stays highlighted across tab switches, so you can browse other categories without losing your place. The tab strip scrolls horizontally on narrow screens.
Selecting an Identity
DemoJump to and the explanation panel
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Two ways to pick an identity. Use the formula buttons under the active tab to choose from identities in that category — each button shows the set-theory notation (A∪B, A′, A△B, and so on). Or use the Jump to dropdown, which lists every identity across all seven categories in one menu, grouped by tab.
When you select an identity, three things update simultaneously:
• The diagram shading changes to highlight the regions belonging to the new identity • The badge above the diagram updates to the new symbol • The explanation panel refreshes with a definition and, where applicable, a numerical example like A={1,2,3,4}, B={3,4,5,6}
Selection is preserved when you switch tabs, so you can compare an identity to others without re-selecting after navigating.
Reading the Shaded Venn Diagram
A two-circle Venn diagram divides the universe into four disjoint regions, and every two-set identity highlights some combination of them:
• Outside both circles — elements not in A and not in B, formally A′∩B′ • Only in A — elements in A but not in B, formally A∖B • Only in B — elements in B but not in A, formally B∖A • In both — the intersection A∩B
For example, A∪B shades the three regions inside either circle. A′ shades the two regions outside circle A (the outside region plus B-only). The symmetric difference A△B shades the two crescent regions but leaves the central overlap unshaded. The complement of the union (A∪B)′ shades only the region outside both circles.
Hover over any region for a tooltip naming it. Identities in the Relations group use special circle layouts (nested, separated, or coincident) instead of the standard overlap.
Customizing Color and Opacity
DemoTheme, Previous and Next
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The Theme panel below the diagram offers two adjustments to the shaded regions.
The color picker changes the shading hue. Useful when printing, presenting, comparing diagrams side by side, or matching the color scheme of a course or textbook. Any standard color value works.
The opacity slider controls how transparent the shading is, ranging from 1.00 (fully opaque) to 0.00 (invisible). Lower opacity is helpful when you want to see the underlying circle outlines through the fill, or when overlaying the diagram on other content. The current numeric value appears next to the slider in monospace.
Click Reset to return both controls to the defaults — blue at 0.85 opacity. Theme changes persist as you navigate between scenarios, so adjustments stay applied across the entire session.
Previous and Next Navigation
At the bottom of the diagram column, the Previous and Next buttons cycle through all 19 identities in the order defined by the category groups: Basic Sets, then Complements, then Intersection & Union, then Differences, Compound, De Morgan's Laws, and finally Relations. The counter between the two buttons displays the current position, formatted as "n / 19".
Navigation wraps around: pressing Previous on the first scenario jumps to the last, and pressing Next on the last returns to the first. This makes the explorer well suited for systematic review — start at the first identity and click through every region one by one to see how each algebraic expression maps to a shaded combination of the four regions.
The active tab and active formula button update automatically as you advance, so you always know which group the current identity belongs to.
The Basic Sets
Before any operation happens, the explorer can display the raw material of two-set algebra: the four atomic states of the Basic Sets tab. Set A on its own and set B on its own each shade one full circle. The universal set shades every region inside the rectangle, and the empty set shades nothing at all.
These four are worth visiting first, because every other identity in the catalog is assembled from them: each of the 19 shadings is just a choice of which of the four regions belongs to A, to B, to both, or to neither. Once you can read the two extremes — everything shaded for U, nothing shaded for ∅ — every intermediate shading becomes a statement about membership.
The Set A on Its Own
The simplest possible selection: the explorer shades everything belonging to A — the left crescent and the central overlap together, making up the full left circle.
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Set A, frozen
The full left circle is shaded: the A-only crescent and the shared lens together. Even the simplest set occupies two of the diagram’s four regions.
The frozen frame makes a point that is easy to miss when sets are only written symbolically: inside a universe with a second set around, A is not one region but two. The diagram decomposes it as
A=(A∖B)∪(A∩B)
and the two pieces are disjoint — an element of A either is in B too, or it is not, never both. This two-piece decomposition is used constantly: it is how counting formulas split ∣A∣, and it is the visual reason the difference A∖B and the intersection recombine into the whole set.
The same picture also illustrates the blandest-looking identities of the algebra, A∩U=A and A∪∅=A: intersecting with everything changes nothing, uniting with nothing changes nothing. Compare the mirror state set B on its own — same shape, opposite side.
The Set B on Its Own
The mirror of the previous state: the shading covers the full right circle — the B-only crescent plus the central overlap.
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Set B, frozen
The mirror image of set A — the full right circle. Swapping the labels turns either frame into the other, which is commutativity drawn as symmetry.
Nothing about the algebra distinguishes the two circles, and the pair of frames proves it: swap the labels A and B and each frozen picture becomes the other. This left-right symmetry of the diagram is the geometric face of commutativity — any identity that survives swapping A and B, like A∪B=B∪A, must shade a left-right symmetric region.
The decomposition works the same way as for A: the set splits as B=(B∖A)∪(A∩B), one part private, one part shared. When you later look at the difference B minus A, it is exactly this private part on its own.
The Universal Set U
Selecting U shades every region — both crescents, the overlap, and the space outside the circles. The whole rectangle lights up.
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Universal set U, frozen
Every region is shaded, crescents, lens and outside alike. The rectangle’s boundary is the only unshaded thing left — the edge of “everything under consideration”.
The universal set is a modeling decision, not a mathematical discovery: it is whatever "everything under consideration" means in the problem at hand — all students in a school, all integers, all outcomes of an experiment. The rectangle in a Venn diagram is that decision drawn as a boundary.
Two algebraic roles follow directly from the all-shaded picture. As the largest possible set, U absorbs unions (A∪U=U) and vanishes from intersections (A∩U=A). And it anchors complementation: A′ only means something relative to a fixed U, and the extreme cases U′=∅ and ∅′=U pair this state with its exact opposite, the empty set.
The Empty Set
Selecting ∅ shades nothing. The frozen frame is the diagram at rest: circles, rectangle, labels — and zero highlighted regions.
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Empty set ∅, frozen
Zero regions shaded. The bare circles are not a missing answer — they are the answer: the picture of a set with no elements.
An all-blank diagram is not a degenerate case; it is a legitimate answer. Plenty of natural expressions evaluate to the empty set — A∩A′ for any A, or the intersection of disjoint sets — and when the explorer produces no shading at all, that is the picture of such an expression.
The empty set's algebra mirrors the universal set's, with the roles reversed: it vanishes from unions (A∪∅=A) and absorbs intersections (A∩∅=∅). One more fact has no visual clue at all and is worth stating: ∅⊆A for every set A — vacuously, since there is no element of ∅ that could fail to be in A.
The Two Complements
The Complements tab holds the two states that flip shading inside-out: the complement of A and the complement of B. Complementation is the only basic operation that shades the region outside both circles, which is why these two states look so different from every union or intersection.
The complement is also the operation that makes the universal set matter. Without a fixed U, "everything not in A" is not a well-defined set; with one, A′ is simply U∖A, and the diagram shows it as the rectangle with a circular hole.
The Complement of A
Selecting A′ turns the shading inside-out: highlighted are the outside region and the B-only crescent — everything except the full circle of A.
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Complement A′, frozen
The rectangle is painted with a circular hole where A was. Note the split through B: its private crescent is shaded, its shared lens is not.
The frozen frame is the rectangle with a circular hole, and the hole is exactly the previous shading of set A on its own. That is the definition made visible: A′=U∖A, so the two states partition the rectangle between them, with no overlap and nothing left over. In symbols, A∪A′=U and A∩A′=∅ — the complement laws, both readable straight off the pair of pictures.
Note that part of B is shaded and part is not: the B-only crescent belongs to A′, but the central overlap does not, because its elements are in A. The complement cuts through B without any regard for it — a first hint of why compound expressions like (A∪B)′ in the De Morgan states need care rather than guesswork.
The Complement of B
The mirror complement: B′ shades the outside region and the A-only crescent — everything except the full circle of B.
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Complement B′, frozen
The negative of set B: outside plus the A-only crescent. Together with B itself it covers the rectangle exactly once.
Together with the complement of A, this frame completes a tidy four-way bookkeeping of the diagram: each of the four regions is either in A or in A′, and independently either in B or in B′. The four combinations A∩B, A∩B′, A′∩B, A′∩B′ are exactly the four regions — overlap, left crescent, right crescent, outside.
Applying the operation twice undoes it: (B′)′=B, the double complement law. In the picture, complementing swaps shaded and unshaded regions, and swapping twice restores the original — which is also why the complement pairs states across the whole catalog, matching each shading with its photographic negative.
Intersection and Union
The two central operations of set algebra get one state each: the intersection shades the single overlap region, and the union shades all three regions inside the circles. Together they are the diagram's translation of the logical words and and or.
They are also each other's extremes: the intersection is the smallest set containing exactly the shared elements, the union the smallest set containing both A and B whole. Every set sandwiched between them — A∩B⊆S⊆A∪B — is expressible from the diagram's regions.
The Intersection of A and B
Selecting A∩B shades a single region: the lens where the two circles overlap. This is the tightest shading of any operation in the catalog — one region out of four.
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Intersection A ∩ B, frozen
One region only — the lens where the circles overlap. The tightest shading any operation produces.
The intersection answers the question and: an element is highlighted exactly when it is in Aand in B. With the explorer's example sets A={1,2,3,4} and B={3,4,5,6}, the lens holds {3,4} — the elements the two sets share.
Two structural facts are visible in the frozen frame. First, A∩B sits inside both circles, which is the picture of A∩B⊆A and A∩B⊆B: the intersection is always a subset of each of its parents. Second, the lens is the region that disjoint sets lose entirely — when it is empty, the sets have nothing in common, and the Relations tab redraws them apart.
The Union of A and B
Selecting A∪B shades three regions at once — both crescents and the overlap — covering everything inside either circle and leaving only the outside blank.
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Union A ∪ B, frozen
Three regions shaded, only the outside left blank. The lens is painted once, not twice — the visual root of inclusion-exclusion.
The union answers the question or, in the inclusive sense: in A, in B, or in both. With A={1,2,3} and B={3,4,5} the union is {1,2,3,4,5} — the shared element 3 appears once, not twice, because sets do not count multiplicity.
The three-region shading explains the most used counting formula in elementary set theory. Adding ∣A∣ and ∣B∣ counts the lens twice — it is inside both circles — so the correction
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣
subtracts the intersection exactly once. The frozen frame is the inclusion-exclusion principle before any formula is written: three disjoint regions, each counted once.
The Three Differences
The Differences tab collects the three subtraction-flavored states: A minus B shades the left crescent, B minus A shades the right crescent, and the symmetric difference shades both crescents at once.
The trio makes a precise algebraic point. Unlike union and intersection, set difference is not commutative — the two one-sided differences shade different regions — and the symmetric difference is exactly the repair: A△B=(A∖B)∪(B∖A) is commutative again, because it treats both crescents equally.
The Difference A Minus B
Selecting A∖B shades only the left crescent: the part of A that does not touch B. The overlap, though it belongs to A, is excluded.
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Difference A ∖ B, frozen
Only the left crescent: A with its shared lens deleted. What remains of A after B takes its part back.
The difference is intersection in disguise: A∖B=A∩B′, "in A and not in B". The frozen frame proves the equivalence — start from the full circle of A and delete the lens, or intersect A with the complement of B; either route paints the same crescent.
Subtraction of sets, like subtraction of numbers, is order-sensitive. The mirror state B minus A shades the other crescent, and the two results share no elements at all — a stronger asymmetry than numbers show, where a−b and b−a at least sit symmetrically around zero. The two crescents together, with the lens still excluded, form the symmetric difference.
The Difference B Minus A
The reversed subtraction: B∖A shades only the right crescent — the private part of B, with the shared lens removed.
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Difference B ∖ A, frozen
The opposite crescent from A ∖ B — same sets, reversed subtraction, disjoint answer. Order matters.
Comparing this frame with A minus B side by side is the cleanest possible demonstration that set difference is not commutative: same two sets, same two circles, opposite crescents, and not a single element in common between the two answers.
The identity B∖A=B∩A′ holds just as its mirror does, and one consequence deserves spelling out: the three sets A∖B, A∩B, and B∖A are pairwise disjoint and together cover A∪B. Every element of the union lives in exactly one of the three — the left crescent, the lens, or the right crescent. That three-way split is the skeleton on which all two-set counting arguments hang.
The Symmetric Difference
Selecting A△B shades both crescents while leaving the central lens blank — the elements belonging to exactly one of the two sets.
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Symmetric difference A △ B, frozen
Both crescents shaded, the lens conspicuously blank: elements of exactly one set. Shared elements cancel.
The symmetric difference is the exclusive or of set theory: in A or in B, but not in both. The frozen frame shows its two equivalent constructions at a glance. Union the two one-sided differences, (A∖B)∪(B∖A), and you paint the two crescents directly; or take the union and delete the intersection, (A∪B)∖(A∩B), and you arrive at the same picture from outside in.
Unlike the one-sided differences, △ is commutative — the shading is left-right symmetric — and it has an algebra of its own: A△A=∅, A△∅=A, and the operation is even associative, making it a group operation on subsets of U. The blank lens is the whole story: shared elements cancel.
These two states are the diagram's version of logical implication. A′∪B is the set form of "if x∈A then x∈B" — it fails only on the A-only crescent, precisely where an element is in A without being in B. Its mirror A∪B′ encodes the reverse implication.
The Union of A with the Complement of B
Selecting A∪B′ shades three regions — outside, the A-only crescent, and the lens — leaving blank exactly one region: the B-only crescent.
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A ∪ B′, frozen
Everything except the B-only crescent. Read it from the blank: the one way out is being in B without A.
Reading compound shadings is easiest through the blank region rather than the shaded ones. Here the unshaded crescent is B∖A, so the frame says A∪B′=(B∖A)′ — the complement of the difference B minus A. One glance at the two frozen frames confirms they are photographic negatives of each other.
The same region-reading gives the logical meaning: an element fails to be in A∪B′ only by being in B without being in A. So the state encodes the implication "if x∈B then x∈A" — it is the truth set of B⇒A. Its mirror, the union of the complement of A with B, encodes the reverse implication.
The Union of the Complement of A with B
The mirror compound: A′∪B shades outside, the B-only crescent, and the lens, leaving only the A-only crescent blank.
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A′ ∪ B, frozen
Everything except the A-only crescent — the truth set of “if in A, then in B”. The blank region is the counterexample zone.
The blank region is A∖B, so this state is the complement of the difference A minus B: A′∪B=(A∖B)′. In logical form it is the truth set of the implication A⇒B — the only way to violate "if in A, then in B" is to sit in the A-only crescent, and that is precisely the one unshaded region.
This state also quietly contains the subset relation. Saying A⊆B is saying the implication holds for every element of U — that is, A′∪B=U, all four regions shaded. Compare A as a subset of B in the Relations tab, where the same fact is drawn by moving the circles instead of shading around them.
What is a Two-Set Venn Diagram?
A two-set Venn diagram is a visual representation of two sets drawn as overlapping circles inside a rectangle. The rectangle represents the universal setU — everything under consideration. The two circles, labeled A and B, represent the two specific sets, and their overlap shows elements common to both.
The diagram has exactly four disjoint regions: outside both circles, only in A, only in B, and the intersection. Every algebraic combination of two sets — no matter how complex — maps to some union of these four regions. This is what makes two-set Venn diagrams a complete visual language for two-set algebra.
Three-set Venn diagrams have eight regions and are harder to read, while one-set diagrams have only two regions and are usually unnecessary. The two-set diagram sits at the practical sweet spot: rich enough to display every standard identity, simple enough to interpret at a glance.
For comprehensive theory on Venn diagrams across different numbers of sets, see Venn diagrams.
Set Operations on Two Sets
Five core operations generate every two-set identity:
UnionA∪B — elements in A, in B, or in both. Set-builder form: {x:x∈A or x∈B}. Visually, the entire shaded region of the two circles.
IntersectionA∩B — elements in both A and B simultaneously. Set-builder form: {x:x∈A and x∈B}. Visually, the overlap of the circles.
ComplementA′ — elements in U but not in A. Set-builder form: {x:x∈/A}. Visually, everything outside circle A.
DifferenceA∖B — elements in A but not in B, equivalent to A∩B′. Visually, the crescent of A that does not overlap B.
Symmetric differenceA△B — elements in exactly one of the two sets, equivalent to (A∖B)∪(B∖A) or (A∪B)∖(A∩B). Visually, both crescents but not the overlap.
For formal definitions and algebraic properties, see set operations.
De Morgan's Laws for Two Sets
De Morgan's laws relate the complement of a combined set to the combination of complements:
(A∪B)′=A′∩B′
(A∩B)′=A′∪B′
The complement of the union equals the intersection of the complements. The complement of the intersection equals the union of the complements. Each law converts a complement of a combination into a combination of complements — useful both for algebraic manipulation and for translating logical statements.
Both laws can be verified visually with the explorer. Select (A∪B)′ from the De Morgan's Laws tab: only the region outside both circles is shaded. That same region is what A′∩B′ would produce — outside A and outside B simultaneously. The two expressions describe the same set, and the diagram confirms it.
For algebraic proofs, the general n-set form, and applications to propositional logic, see De Morgan's laws.
Selecting (A∪B)′ shades a single region: the space outside both circles. Nothing inside either circle is highlighted.
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(A ∪ B)′, frozen
A single shaded region: outside both circles. Outside the union means outside A and outside B at once — De Morgan’s first law as a picture.
This is the most extreme shading in the catalog — one region, and the one no basic operation reaches on its own. The frame is the visual proof of the first De Morgan law: being outside the union means being outside Aand outside B simultaneously, which is precisely the intersection of the two complements,
(A∪B)′=A′∩B′
You can verify it with the earlier frames: overlay the complement of A and the complement of B and keep only regions shaded in both — the B-only crescent falls (not in A′'s partner), the A-only crescent falls, and only the outside survives. Complement turned a union into an intersection; that swap is the whole content of the law.
The Complement of the Intersection
Selecting (A∩B)′ shades three regions — outside and both crescents — leaving only the central lens blank.
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(A ∩ B)′, frozen
Three regions shaded, only the lens spared. Avoiding the intersection means missing at least one set — the second law.
The second De Morgan law is this frame's caption written in symbols: avoiding the intersection means missing at least one of the two sets, i.e. being outside Aor outside B,
(A∩B)′=A′∪B′
Note the mirrored structure with the complement of the union: there, complement turned ∪ into ∩ and shaded 1 region; here it turns ∩ into ∪ and shades 3. The two shadings are complementary — together they cover all four regions, overlapping nowhere, because their unpainted parts (the union and the intersection respectively) nest inside one another.
A useful reading habit generalizes from here: a complemented compound expression is easiest to shade by first shading the inside expression, then flipping every region. The explorer lets you do exactly that experiment with one click.
Set Relations
The Relations tab is different in kind from every other tab: instead of shading regions of the standard overlapping layout, it redraws the circles to express a relationship between the sets. A as a subset of B nests a small circle inside a large one, B as a subset of A mirrors it, disjoint sets pulls the circles apart, and equal sets draws them on top of each other.
This is an honest limitation of Venn diagrams worth understanding: the standard two-circle layout deliberately shows the general position, where all four regions are possible. A relation like A⊆B asserts that one of those regions is empty, and the clearest way to draw an empty region is not to draw it at all.
A as a Subset of B
Selecting A⊆B redraws the layout: a small circle A sits entirely inside a large circle B, and the shading fills A — which is now the same thing as the overlap A∩B.
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A ⊆ B, frozen
The layout itself changes: A nests inside B, so an A-only region cannot even be drawn. The shaded A is simultaneously the whole intersection.
The nested layout encodes the relation geometrically: there is nowhere to stand inside A without also standing inside B, so the A-only crescent of the standard layout has vanished. That vanished region is the assertion — A⊆B is equivalent to A∖B=∅.
Two algebraic consequences become obvious in the picture: A∩B=A (the overlap is all of A) and A∪B=B (uniting adds nothing new to B). Either equation can serve as the definition of subset, and both degenerate gracefully in the extremes: ∅⊆B for any B, and B⊆B always — the subset relation is reflexive. When it holds in both directions at once, the sets are equal; see equal sets for that limiting layout, and B as a subset of A for this state's mirror.
B as a Subset of A
The mirrored nesting: a small circle B inside a large circle A, with the shading filling B — again the entire overlap of the pair.
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B ⊆ A, frozen
The mirrored nesting: B sits wholly inside A. Here A ∩ B = B and A ∪ B = A, both visible at a glance.
Everything from A as a subset of B transfers with the roles reversed: B∖A=∅, A∩B=B, and A∪B=A. The pair of nested frames also makes antisymmetry visible — if each set is a subset of the other, neither circle can be strictly smaller, and the layouts collapse into the coincident circles of equal sets.
One caution the drawing teaches by contrast: in the standard overlapping layout, nothing stops B⊆A from being true — the picture merely fails to show it, since it draws a B-only crescent that would in fact be empty. The Relations layouts exist precisely to remove that silent possibility from the page.
Disjoint Sets
Selecting A∩B=∅ separates the circles completely: two non-overlapping discs, both shaded, with clear space between them.
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Disjoint sets, frozen
The circles separate entirely — no lens exists to shade. With nothing shared, the union is a plain side-by-side placement.
Disjointness is the empty-intersection relation: the sets share no elements, so the lens region of the standard layout is gone and the circles need not touch. The shading here highlights both discs — everything that is in A or in B — showing that for disjoint sets the union simply places the two sets side by side.
Counting is where disjointness pays off: with no shared elements to double-count, inclusion-exclusion loses its correction term and ∣A∪B∣=∣A∣+∣B∣ exactly. This is the additivity that underlies probability of mutually exclusive events and every "partition into cases" argument. Note the contrast with the intersection state in the standard layout, whose single shaded lens is exactly what disjointness declares empty.
Equal Sets
Selecting A=B draws the two circles in the same position — one solid outline, one dashed so both remain visible — and shades the single shared disc.
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Equal sets, frozen
Two circles drawn in one place, the second dashed so it stays visible. All the interior structure of the diagram has collapsed into a single shared disc.
Equality is the degenerate limit of the diagram: all the structure that makes a Venn diagram interesting has collapsed. There is no A-only crescent, no B-only crescent — just one region inside and one outside. The dashed second outline is a drawing trick, not mathematics: coincident circles would otherwise be indistinguishable from a single set.
The working definition behind the picture is double inclusion: A=B exactly when A⊆B and B⊆A. That is how equality of sets is actually proved — two subset arguments, one in each direction — and the two nested layouts of A as a subset of B and B as a subset of A are the two halves of the proof, superimposed here into a single coincident frame. In element terms: sets are equal when they have exactly the same members, regardless of how they are described.
Related Concepts and Tools
Set Operations — formal definitions and properties of union, intersection, complement, difference, and symmetric difference.
Venn Diagrams — overview of one-set, two-set, and three-set diagrams, drawing conventions, and when each is appropriate.
De Morgan's Laws — algebraic proofs of both two-set laws and the generalization to arbitrary collections of sets.
Set Relations — definitions of subset, proper subset, disjoint sets, and equal sets, with examples.
Set Theory Definitions — glossary of foundational terms used throughout set algebra.
Three-Set Venn Diagram — extends the same visual approach to three overlapping sets and the eight regions they produce, including the three-set De Morgan's laws.
Set Laws and Identities — algebraic catalog of commutative, associative, distributive, absorption, and complement laws on sets.