Union collects the elements in either set; intersection keeps the elements in both; difference keeps those in one set but not the other; the complement is everything in U outside a set.
An operation on numbers, such as addition, takes two numbers and gives a number. An operation on sets takes sets and gives a set. Four operations are used here. Each corresponds to a region of a Venn diagram.
The examples use A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.
The union of A and B, written A ∪ B, is the set of elements that are in A or in B or in both:
A ∪ B = { x : x ∈ A or x ∈ B } = {1, 2, 3, 4, 5, 6}
3 and 4 are in both sets but are written once. In a Venn diagram, the union is the whole of both circles.
The intersection of A and B, written A ∩ B, is the set of elements that are in both A and B:
A ∩ B = { x : x ∈ A and x ∈ B } = {3, 4}
In a Venn diagram, it is the overlap of the two circles. If A ∩ B = ∅, the sets have no element in common and are called disjoint.
For A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, compare n(A ∪ B) and n(A ∩ B) with n(A) and n(B).
n(A) = 4 and n(B) = 4. A ∪ B = {1, 2, 3, 4, 5, 6}, so n(A ∪ B) = 6. A ∩ B = {3, 4}, so n(A ∩ B) = 2.
Every element of A is in A ∪ B, so the union has at least as many elements as either set. Every element of A ∩ B is in A, so the intersection has at most as many elements as either set.
Also: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 4 + 4 − 2 = 6. Adding n(A) and n(B) counts the shared elements twice, so they are subtracted once.
The difference A − B is the set of elements of A that are not in B:
A − B = {1, 2} B − A = {5, 6}
In a Venn diagram, A − B is the part of circle A outside the overlap. The order matters: A − B and B − A are different sets.
The complement of A, written A′, is the set of elements of the universal set U that are not in A. So A′ = U − A. It is the only one of the four operations that depends on U.
If U = {1, 2, …, 10} and A = {1, 2, 3, 4}, then A′ = {5, 6, 7, 8, 9, 10}. In a Venn diagram, A′ is the part of the rectangle outside circle A.
With U = {1, 2, …, 10} and A = {2, 4, 6, 8, 10}, what is A′?
A′ is every element of U not in A: {1, 3, 5, 7, 9}, the odd numbers from 1 to 10.
With this U, the complement of the even numbers is the odd numbers. With U = {1, 2, …, 20}, A′ would also contain 11, 12, 13 and so on up to 20, because only 2 to 10 are in A.
A set is a well-defined collection, written in roster or set-builder form. Sets may be empty, finite or infinite, and are equal when they have the same elements. A subset lies inside another set; a set with n elements has 2ⁿ subsets; intervals are subsets of the real line. The universal set fixes the elements under discussion, and Venn diagrams show sets as regions. Union, intersection, difference and complement build new sets from old ones, and each corresponds to a region of a Venn diagram.
1A ∪ B is the set of elements that are…
2For A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, A ∩ B is…
3Two sets with A ∩ B = ∅ are called…
4For A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, A − B is…
5Why does the complement A′ depend on the universal set?
6With U = {1, 2, …, 10} and A = {2, 4, 6, 8, 10}, A′ = ____.
7Which is always true for any two sets A and B?
That completes the unit on sets.
Noescio