# Intersection

In set theory, the intersection of a collection of sets is the set that contains their shared elements. Given two sets, A = {2, 3, 4, 7, 10} and B = {1, 3, 5, 7, 9}, their intersection is as follows:

A ∩ B = {3, 7}

The intersection of two sets is commonly represented using a Venn diagram. In a Venn diagram, a set is represented by a circle. The intersection of sets A and B in the Venn diagram below is the shaded region where the two circles overlap.

A = {1, 2, 3, 4, 5,} and B = {4, 5, 6, 7, 8, 9, 10}, so A ∩ B = {4, 5}. Note that A and B are part of 𝕌, the universal set (represented by the rectangle), which contains other elements that are not part of A or B.

## Properties of intersections

Like other basic operations such as addition, set operations like intersections also have certain properties. Refer to the set page if necessary for a table of symbols commonly used in set theory.

### Intersections and subsets

If set A is a subset of set B, then the intersection of the two sets is equal to set A. Using set notation:

if A ⊆ B, then A ∩ B = A

For example, if A = {4, 5, 6} and B = {4, 5, 6, 7, 8}, their intersection is {4, 5, 6}, or A.

### Commutative law

The commutative law states that the order in which the intersection of two sets is taken does not matter. Given two sets, A and B:

A ∩ B = B ∩ A

Let A = {1, 2, 3} and B = {3, 5, 7}. The only common element the two sets have is 3. Thus, A ∩ B = B ∩ A = {3}. It doesn't matter whether we consider A or B first, the result will be the same.

### Associative law

The associative law states that rearranging the parentheses in an intersection of sets does not change the result. Given sets A, B, and C:

(A ∩ B) ∩ C = A ∩ (B ∩ C)

The position of the parentheses doesn't matter since the intersection operation does not change the relationship between the sets. Comparing sets A and B first, then B and C, or B and C first, then A and B, doesn't change the relationship between A and B or B and C.

### Distributive law

For sets A, B, and C, the distributive law states

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

where A is distributed to B and C. This is similar to the distributive property of multiplication in which multiplication distributes over addition. The Venn diagram below demonstrates the first law listed above:

From the figure, A = {1, 2, 3, 4, 5, 6}, B = {3, 4, 5, 7, 8, 9, 10}, and C = {4, 5, 6, 7, 8, 11, 12}. Thus:

B ∩ C = {4, 5, 7, 8}

A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 11, 12}

The left side of the equation is:

A ∪ (B ∩ C) | = | {1, 2, 3, 4, 6, 6} ∪ {4, 5, 7, 8} |

= | {1, 2, 3, 4, 5, 6, 7, 8} |

The right side of the equation is:

(A ∪ B) ∩ (A ∪ C) | = | {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ∩ {1, 2, 3, 4, 5, 6, 7, 8, 11, 12} |

= | {1, 2, 3, 4, 5, 6, 7, 8} |

The left and right sides are equal, confirming the distributive law.

### De Morgan's laws

In set theory, De Morgan's laws are a set of rules that relate the union and intersection of sets through their complements.

#### Intersection of sets:

The complement of the intersection of two sets is equal to the union of their complements:

A ∩ B = A^{C} ∪ B^{C}

Given that A and B are subsets of the universal set 𝕌, this relationship can be seen in the figure below:

The intersection of A and B, A ∩ B, is shaded in red. Its complement, (A ∩ B)^{C} is shaded in grey. The union of the complements of A and B, A^{C} ∪ B^{C}, is also shaded in grey.

#### Union of sets:

The complement of the union of two sets is equal to the intersection of their complements:

(A ∪ B)^{C} = A^{C} ∩ B^{C}

Given that A and B are subsets of the universal set 𝕌, this relationship can be seen in the figure below:

The union of A and B, A ∪ B, is shaded in blue. Its complement, (A ∪ B)^{C} is shaded in yellow. The intersection of the complements of A and B, A^{C} ∩ B^{C} is also shaded in yellow.