Set Operations

Union Of Sets

Formula:

A \cup B = \{ x \mid x \in A \ \text{or} \ x \in B \}

Explanation

The union of two sets brings together all the elements from both sets without any duplicates. Think of it as combining two groups into one bigger group that contains every unique member from both.

A

B

The sets being united

x

An element

x \in A

Element

x

is in set

A

x \in B

Element

x

is in set

B

Example

A = \{1, 2, 3\}

and

B = \{3, 4, 5\}

, then

A \cup B = \{1, 2, 3, 4, 5\}

Use Cases

Combining data sets, merging lists, unifying groups

Intersection Of Sets

Formula:

A \cap B = \{ x \mid x \in A \ \text{and} \ x \in B \}

Explanation

The intersection of two sets includes only the elements that are present in both sets. It's like finding common friends between two people.

A

B

The sets being intersected

x

An element

x \in A

and

x \in B

Element

x

is in both sets

A

and

B

Example

A = \{2, 3, 4\}

and

B = \{3, 4, 5\}

, then

A \cap B = \{3, 4\}

Use Cases

Finding commonalities, overlapping interests, shared attributes

Set Difference

Formula:

A \setminus B = \{ x \mid x \in A \ \text{and} \ x \notin B \}

Explanation

The difference between two sets

A

and

B

consists of elements that are in

A

but not in

B

. It's like removing certain items from your collection.

A

B

The sets involved

x

An element

x \in A

Element

x

is in set

A

x \notin B

Element

x

is not in set

B

Example

A = \{1, 2, 3, 4\}

and

B = \{3, 4, 5\}

, then

A \setminus B = \{1, 2\}

Use Cases

Excluding items, filtering data, subtracting groups

Complement Of A Set

Formula:

A' = U \setminus A = \{ x \mid x \in U \ \text{and} \ x \notin A \}

Explanation

The complement of a set includes everything that's not in the set, relative to a universal set

U

. It's like considering all the things you're not choosing.

A

The set you're taking the complement of

U

The universal set containing all possible elements

x

An element

x \notin A

Element

x

is not in set

A

x \in U

Element

x

is in the universal set

U

Example

U = \{1, 2, 3, 4, 5\}

and

A = \{2, 3\}

, then

A' = \{1, 4, 5\}

Use Cases

Finding what is excluded, logical negation, probability of the opposite event

Cartesian Product

Formula:

A \times B = \{ (a, b) \mid a \in A, \ b \in B \}

Explanation

The Cartesian product pairs every element of set

A

with every element of set

B

, forming ordered pairs. Imagine creating all possible combinations between two lists.

A

B

The sets being combined

(a, b)

An ordered pair with

a

from

A

and

b

from

B

a \in A

Element

a

is in set

A

b \in B

Element

b

is in set

B

Example

A = \{1, 2\}

and

B = \{x, y\}

, then

A \times B = \{ (1, x), (1, y), (2, x), (2, y) \}

Use Cases

Creating coordinate grids, modeling relationships, forming combinations

Fundamental Properties

Commutative Law Of Union

Formula:

A \cup B = B \cup A

Explanation

The order in which you unite two sets doesn't matter; you'll end up with the same set. It's like mixing two ingredients together—the result is the same no matter the order.

A

B

The sets being united

Example

A = \{apple, banana\}

and

B = \{banana, cherry\}

, then

A \cup B = \{apple, banana, cherry\} = B \cup A

Use Cases

Simplifying set expressions, proving set identities

Commutative Law Of Intersection

Formula:

A \cap B = B \cap A

Explanation

The order in which you find the common elements between two sets doesn't matter. It's like checking shared interests between two people; the overlap is the same regardless of who you start with.

A

B

The sets being intersected

Example

A = \{red, blue\}

and

B = \{blue, green\}

, then

A \cap B = \{blue\} = B \cap A

Use Cases

Simplifying calculations, proving properties of sets

Associative Law Of Union

Formula:

(A \cup B) \cup C = A \cup (B \cup C)

Explanation

When uniting multiple sets, the grouping doesn't affect the result. It's like grouping ingredients differently when mixing; the final mix is the same.

A

B

C

The sets being united

Example

A = \{1\}

B = \{2\}

C = \{3\}

, then

(A \cup B) \cup C = A \cup (B \cup C) = \{1, 2, 3\}

Use Cases

Rearranging expressions for simplification, computational efficiency

Associative Law Of Intersection

Formula:

(A \cap B) \cap C = A \cap (B \cap C)

Explanation

When finding common elements among multiple sets, how you group them doesn't change the outcome. It's like narrowing down shared interests among friends; the sequence doesn't matter.

A

B

C

The sets being intersected

Example

A = \{1,2\}

B = \{2,3\}

C = \{2,4\}

, then

(A \cap B) \cap C = A \cap (B \cap C) = \{2\}

Use Cases

Simplifying complex intersections, logical deductions

Distributive Law Of Intersection Over Union

Formula:

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Explanation

Intersecting a set with the union of two sets is the same as uniting the intersections of the set with each of the two sets individually. It's like finding common items after combining two lists separately.

A

B

C

The sets involved

Example

A = \{1,2,3\}

B = \{3,4\}

C = \{3,5\}

, then

A \cap (B \cup C) = \{3\}

and

(A \cap B) \cup (A \cap C) = \{3\}

Use Cases

Simplifying expressions, solving set equations

Distributive Law Of Union Over Intersection

Formula:

A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

Explanation

Uniting a set with the intersection of two sets is the same as intersecting the unions of the set with each of the two sets individually. It's like combining your list with shared items of others, or finding common ground after including all possibilities.

A

B

C

The sets involved

Example

A = \{1\}

B = \{1,2\}

C = \{1,3\}

, then

A \cup (B \cap C) = \{1\}

and

(A \cup B) \cap (A \cup C) = \{1,2,3\} \cap \{1,3\} = \{1,3\}

Use Cases

Rewriting expressions for easier computation, logical reasoning

Identity Law Of Union

Formula:

A \cup \emptyset = A

Explanation

Uniting any set with the empty set doesn't change the original set. It's like adding nothing to a group; the group stays the same.

A

Any set

\emptyset

The empty set

Example

A = \{1,2\}

, then

A \cup \emptyset = \{1,2\}

Use Cases

Simplifying expressions, understanding the role of the empty set

Identity Law Of Intersection

Formula:

A \cap U = A

Explanation

Intersecting any set with the universal set leaves the original set unchanged. It's like comparing your collection to everything possible; only your items matter.

A

Any set

U

The universal set

Example

A = \{1,2\}

and

U

contains all numbers, then

A \cap U = \{1,2\}

Use Cases

Simplifying expressions, theoretical foundations

Idempotent Law Of Union

Formula:

A \cup A = A

Explanation

Uniting a set with itself doesn't change it. It's like combining a group with itself; nothing new is added.

A

Any set

Example

A = \{1,2,3\}

, then

A \cup A = \{1,2,3\}

Use Cases

Simplifying redundant operations, mathematical proofs

Idempotent Law Of Intersection

Formula:

A \cap A = A

Explanation

Intersecting a set with itself leaves it unchanged. It's like finding common elements within the same group; everything matches.

A

Any set

Example

A = \{x,y,z\}

, then

A \cap A = \{x,y,z\}

Use Cases

Simplifying expressions, reducing complexity in proofs

Double Complement Law

Formula:

(A')' = A

Explanation

Taking the complement of a complement brings you back to the original set. It's like undoing a reversal; you return to where you started.

A

Any set

A'

Complement of set

A

(A')'

Complement of the complement of

A

Example

A = \{1,2\}

within

U = \{1,2,3,4\}

, then

A' = \{3,4\}

and

(A')' = \{1,2\} = A

Use Cases

Simplifying complex expressions, logical negations

Complement Of Universal Set

Formula:

U' = \emptyset

Explanation

The complement of the universal set is the empty set because there are no elements outside of everything considered.

U

The universal set

U'

Complement of the universal set

\emptyset

The empty set

Example

U = \{all\ elements\}

, then

U' = \emptyset

Use Cases

Understanding limits, theoretical concepts

Complement Of Empty Set

Formula:

\emptyset' = U

Explanation

The complement of the empty set is the universal set because it contains everything that isn't in the empty set—which is everything.

\emptyset

The empty set

U

The universal set

\emptyset'

Complement of the empty set

Example

U = \{1,2,3\}

, then

\emptyset' = \{1,2,3\}

Use Cases

Establishing foundational concepts, simplifying expressions

De Morgan's First Law

Formula:

(A \cup B)' = A' \cap B'

Explanation

The complement of the union of two sets is the intersection of their complements. It's like saying everything that's not in either set is what's not in

A

and not in

B

A

B

Sets involved

A'

B'

Complements of

A

and

B

Example

U = \{1,2,3,4\}

A = \{1,2\}

B = \{2,3\}

, then

(A \cup B)' = \{4\}

and

A' \cap B' = \{4\}

Use Cases

Logical reasoning, simplifying complex expressions

De Morgan's Second Law

Formula:

(A \cap B)' = A' \cup B'

Explanation

The complement of the intersection of two sets is the union of their complements. It's like everything not shared between

A

and

B

is everything not in

A

or not in

B

A

B

Sets involved

A'

B'

Complements of

A

and

B

Example

U = \{1,2,3,4\}

A = \{1,2\}

B = \{2,3\}

, then

(A \cap B)' = \{1,3,4\}

and

A' \cup B' = \{1,3,4\}

Use Cases

Logical negations, simplifying expressions in probability and logic

Cardinality

Cardinality Of A Set

Formula:

|A| = n

Explanation

The cardinality of a set is the number of elements it contains. It's like counting how many items are in your collection.

A

The set

|A|

Number of elements in set

A

n

A non-negative integer

Example

A = \{a, b, c\}

, then

|A| = 3

Use Cases

Measuring set sizes, comparing quantities, finite mathematics

Cardinality Of Union Of Two Sets

Formula:

|A \cup B| = |A| + |B| - |A \cap B|

Explanation

When combining two sets, the total number of unique elements is the sum of their sizes minus the number of elements they share. This avoids double-counting shared elements.

A

B

Sets being united

|A|

|B|

Cardinalities of sets

A

and

B

|A \cap B|

Number of elements common to both sets

Example

|A| = 5

|B| = 7

, and

|A \cap B| = 2

, then

|A \cup B| = 5 + 7 - 2 = 10

Use Cases

Calculating probabilities, combining datasets, survey analysis

Principle Of Inclusion-Exclusion For Three Sets

Formula:

|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|

Explanation

To find the total number of unique elements across three sets, we add their sizes, subtract the sizes of all pairwise intersections to correct for double-counting, and then add back the size of the triple intersection to correct for elements subtracted too many times.

A

B

C

Sets being united

|A|

|B|

|C|

Cardinalities of the sets

|A \cap B|

, etc.

Sizes of pairwise intersections

|A \cap B \cap C|

Size of the intersection of all three sets

Example

|A| = 10

|B| = 15

|C| = 20

|A \cap B| = 5

|A \cap C| = 4

|B \cap C| = 6

, and

|A \cap B \cap C| = 2

, then

|A \cup B \cup C| = 10 + 15 + 20 - 5 - 4 - 6 + 2 = 32

Use Cases

Complex probability calculations, overlapping group analysis, database queries

Power Set

Cardinality Of Power Set

Formula:

|\mathcal{P}(A)| = 2^{|A|}

Explanation

The power set of

A

includes all possible subsets of

A

, and its size is

2

raised to the number of elements in

A

. It's like calculating all possible combinations of items you can select from your collection.

A

The original set

\mathcal{P}(A)

Power set of

A

|A|

Number of elements in set

A

2^{|A|}

Total number of subsets

Example

A = \{1,2,3\}

, then

|\mathcal{P}(A)| = 2^3 = 8

Use Cases

Combinatorics, probability, binary representations

Important Rules

Transitivity Of Subsets

Formula:

A \subseteq B

and

B \subseteq C

, then

A \subseteq C

Explanation

If every element of

A

is in

B

, and every element of

B

is in

C

, then every element of

A

must also be in

C

. It's like a chain of containment.

A

B

C

Sets involved

\subseteq

Symbol meaning 'is a subset of'

Example

A = \{1\}

B = \{1,2\}

, and

C = \{1,2,3\}

, then

A \subseteq C

Use Cases

Logical deductions, hierarchical classifications

Empty Set Is A Subset Of Every Set

Formula:

\emptyset \subseteq A

Explanation

The empty set is considered a subset of any set because there are no elements in it that could possibly violate the subset condition. It's like an empty container fitting into any space.

\emptyset

The empty set

A

Any set

\subseteq

Symbol meaning 'is a subset of'

Example

For any set

A

\emptyset \subseteq A

Use Cases

Mathematical proofs, defining foundational concepts

Extensionality Principle

Formula:

A = B \iff \forall x (x \in A \iff x \in B)

Explanation

Two sets are equal if and only if they have exactly the same elements. It's like saying two collections are the same if everything in one is also in the other and vice versa.

A

B

Sets being compared

\forall x

For all elements

x

\iff

If and only if

x \in A

Element

x

is in set

A

Example

A = \{1,2,3\}

and

B = \{3,2,1\}

, then

A = B

Use Cases

Determining set equality, logical reasoning

Set Theory Formulas

Set Operations

Union Of Sets

Formula:

Intersection Of Sets

Formula:

Set Difference

Formula:

Complement Of A Set

Formula:

Cartesian Product

Formula:

Fundamental Properties

Commutative Law Of Union

Formula:

Commutative Law Of Intersection

Formula:

Associative Law Of Union

Formula:

Associative Law Of Intersection

Formula:

Distributive Law Of Intersection Over Union

Formula:

Distributive Law Of Union Over Intersection

Formula:

Identity Law Of Union

Formula:

Identity Law Of Intersection

Formula:

Idempotent Law Of Union

Formula:

Idempotent Law Of Intersection

Formula:

Double Complement Law

Formula:

Complement Of Universal Set

Formula:

Complement Of Empty Set

Formula:

De Morgan's First Law

Formula:

De Morgan's Second Law

Formula:

Cardinality

Cardinality Of A Set

Formula:

Cardinality Of Union Of Two Sets

Formula:

Principle Of Inclusion-Exclusion For Three Sets

Formula:

Power Set

Cardinality Of Power Set

Formula:

Important Rules

Transitivity Of Subsets

Formula:

Empty Set Is A Subset Of Every Set

Formula:

Extensionality Principle

Formula: