CS202: Discrete Structures

1.2 Basic Set Operations

1.2.1 Definitions

Definition 1.2.1: Intersection. Let A and B be sets. The intersection of A and B (denoted by A ∩ B) is the set of all elements that are in both A and B. That is, A ∩ B = {x : x ∈ A and x ∈ B}.

Example 1.2.2: Some Intersections.

• Let A = {1, 3, 8} and B = {−9 , 22, 3}. Then A ∩ B = {3}.
• Solving a system of simultaneous equations such as x+y = 7 and x −y = 3 can be viewed as an intersection. Let A = {(x, y) : x + y = 7, x, y ∈ R } and B = {(x, y ) : x − y = 3, x, y ∈ R}. These two sets are lines in the plane and their intersection, A ∩ B = { (5, 2)}, is the solution to the system.
• Z ∩ Q = Z.
• If A = {3, 5, 9} and B = {−5, 8}, then A ∩ B = ∅.

Definition 1.2.3: Disjoint Sets. Two sets are disjoint if they have no elements in common. That is, A and B are disjoint if A ∩ B = ∅.

Definition 1.2.4: Union. Let A and B be sets. The union of A and B (denoted by A ∪ B) is the set of all elements that are in A or in B or in both A and B. That is, A ∪ B = {x : x ∈ A or x ∈ B}.

It is important to note in the set-builder notation for A ∪ B, the word "or" is used in the inclusive sense; it includes the case where x is in both A and B.

Example 1.2.5: Some Unions.

• If A = {2, 5, 8} and B = {7, 5, 22}, then A ∪ B = {2, 5, 8, 7, 22}.
• Z ∪ Q = Q.
• A ∪ ∅ = A for any set A .

Frequently, when doing mathematics, we need to establish a universe or set of elements under discussion. For example, the set A = {x : 81 x⁴ − 16 = 0} contains different elements depending on what kinds of numbers we allow ourselves to use in solving the equation 81x⁴ − 16 = 0. This set of numbers would be our universe. For example, if the universe is the integers, then A is empty. If our universe is the rational numbers, then A is {2/3, −2/3} and if the universe is the complex numbers, then A is {2/3, − 2/3, 2i/3 , −2i/3}.

Definition 1.2.6: Universe. The universe, or universal set, is the set of all elements under discussion for possible membership in a set. We normally reserve the letter U for a universe in general discussions.

1.2.2 Set Operations and their Venn Diagrams

When working with sets, as in other branches of mathematics, it is often quite useful to be able to draw a picture or diagram of the situation under consideration. A diagram of a set is called a Venn diagram. The universal set U is represented by the interior of a rectangle and the sets by disks inside the rectangle.

Example 1.2.7: Venn Diagram Examples. A ∩ B is illustrated in Figure 1.2.8 by shading the appropriate region.

Figure 1.2.8 Venn Diagram for the Intersection of Two Sets

The union A ∪ B is illustrated in Figure 1.2.9.

Figure 1.2.9 Venn Diagram for the Union A ∪ B

In a Venn diagram, the region representing A ∩ B does not appear empty; however, in some instances it will represent the empty set. The same is true for any other region in a Venn diagram.

Definition 1.2.10: Complement of a set. Let A and B be sets. The complement of A relative to B (notation B − A) is the set of elements that are in B and not in A. That is, B − A = {x : x ∈ B and x ∉ A}. If U is the universal set, then U − A is denoted by A^c and is called simply the complement of A. A^c = { x ∈ U : x ∉ A}.

Figure 1.2.11 Venn Diagram for B − A

Example 1.2.12: Some Complements.

(a) Let U = {1, 2, 3, ..., 10} and A = {2, 4, 6, 8, 10}. Then U − A = {1, 3, 5, 7, 9} and A − U = ∅.

(b) If U = R, then the complement of the set of rational numbers is the set of irrational numbers.

(c) U^c = ∅ and ∅^c = U .

(d) The Venn diagram of B − A is represented in Figure 1.2.11.

(e) The Venn diagram of A^c is represented in Figure 1.2.13.

(f) If B ⊆ A, then the Venn diagram of A − B is as shown in Figure 1.2.14.

(g) In the universe of integers, the set of even integers, {. . . , −4, −2, 0, 2, 4, . . .}, has the set of odd integers as its complement.

Figure 1.2.13 Venn Diagram for A^c