What are the four operations of sets?

Answer 1

The four basic operations of sets are unions, intersections, complements, and the Cartesian product.

Unions

A union is essentially the act of 'adding' multiple sets together to combine their elements into a single set.

example: if A={1,3,5} and B={2,4,6} then the union A∪B={1,2,3,4,5,6} however, the same elements are not counted twice so if A={1,2,5} and B={1,2,4} then A∪B={1,2,4,5}

Intersections

An intersection makes a new set from the common elements of the sets involved.

example: for A={2,3,5,7,11}, B={2,5,8} and C={5,9,11} then the intersection A∩(B∩C)={5}

Notice that even though A and B have 2, and A and C have 11, the intersection of the three sets is 5 as that is the only common element between all three.

Also notice that if you take the intersection of two sets with no elements in common you end up with the empty set.

Complements

There are 2 complements, the relative complement and the absolute complement.

The relative complement is the 'subtraction' of multiple sets. The relative complement of A in B, written B\A, is the set of all elements that are in B but aren't in A.

example: A={1,2,3,4,5,6}, B={5,6,7,8,9} then B\A={7,8,9}

This property does not commute, B\A≠A\B

Given a universal set U, defined as containing all the elements in that area, the absolute complement is the complement of A in U, and is denoted as Ac. i.e. Ac is the set of all elements in U that aren't in A.

example: let U={1,2,3,4,5,6,7,8,9,10}, and A={2,3,5,7}. Then Ac={1,4,6,8,9,10}

bigger example: if U={x∈ℕ} (the set of all positive integers not including 0) and A={x=2k|k∈ℕ} (the set off all positive and even numbers) then Ac is the set of all positive and odd integers.

Cartesian Product

The cartesian product is the combination of elements from multiple sets.

example: Let A={1,2,3} and B={red, blue} then the Cartesian product AxB={(1,red), (1,blue), (2,red), (2,blue), (3,red), (3,blue)}

This property generally does not commute, AxB=BxA if and only if A=B.