Operations on Sets
21
Example 5.
(a) Let A ={1, 2.
10} and B = {3, 5, 7, 9}. Then, A - B = {1, 2, 4, 6, 8, 101.
(b) Let A =N and B = {2i i E NJ. Then, A - B = {2i + 1: i E N}.
The difference A - B is also sometimes call the relative difference. The Venn diagram (shown in Figure 1.9) gives an intuitive understanding of this notion. Remember that
Venn diagrams suggest relations between or among sets but are not actually proofs of relationships between or among sets. Theorem 6 proves some key relationships involving the
difference of two sets, A - B and B - A.
Theorem 6. Let A and B be sets. Then:
(a) A - B and B - A are disjoint, A - B and A n B are disjoint, and A n B and B - A
are disjoint.
(b) A=(A--B) U(ANB).
(c) AUB=(A-B) U(ANB) U (B - A).
(d) ACBifandonlyifA--B=0.
Proof. If you look at a Venn diagram for two sets and identify A - B, B - A, and A n B,
it looks like the sets are disjoint. This theorem says that your intuition from the diagram is
correct. The proofs of (a)-(d) are left as exercises for the reader.
U
Complement of a Set
Recall that a universal set is a set that contains as a subset every set currently being discussed. In a context in which there is a universal set, another set theoretic operation can be
defined.
Definition 6. Let U be a universal set and A be a subset of U. The complement of A,
denoted A, is
{x : x E U andx ý A}
Sometimes, to emphasize that U is a universal set, A is also called the absolute difference.
With this definition, we can restate Definition 5 as A - B = A nf B. Some important
identities concern complements, especially how they interact with other set-theoretic operations.
Theorem 7. Let U be a universal set and A and B be subsets of U. Then:
(a) A = A. (A is the complement of A.)
(b) A C B if and only if B C A.
(c) A = B if and only if A = B.
(What the proof entails.) Part (a) tells us that the complement only produces something
new the first time it is applied. Part (b) says that if A is a subset of B, then set inclusion
goes the other way for the complements; that is, the complement of B is contained in the
complement of A. In part (c), we prove that if two sets are equal, then their complements
are equal.
21
Example 5.
(a) Let A ={1, 2.
10} and B = {3, 5, 7, 9}. Then, A - B = {1, 2, 4, 6, 8, 101.
(b) Let A =N and B = {2i i E NJ. Then, A - B = {2i + 1: i E N}.
The difference A - B is also sometimes call the relative difference. The Venn diagram (shown in Figure 1.9) gives an intuitive understanding of this notion. Remember that
Venn diagrams suggest relations between or among sets but are not actually proofs of relationships between or among sets. Theorem 6 proves some key relationships involving the
difference of two sets, A - B and B - A.
Theorem 6. Let A and B be sets. Then:
(a) A - B and B - A are disjoint, A - B and A n B are disjoint, and A n B and B - A
are disjoint.
(b) A=(A--B) U(ANB).
(c) AUB=(A-B) U(ANB) U (B - A).
(d) ACBifandonlyifA--B=0.
Proof. If you look at a Venn diagram for two sets and identify A - B, B - A, and A n B,
it looks like the sets are disjoint. This theorem says that your intuition from the diagram is
correct. The proofs of (a)-(d) are left as exercises for the reader.
U
Complement of a Set
Recall that a universal set is a set that contains as a subset every set currently being discussed. In a context in which there is a universal set, another set theoretic operation can be
defined.
Definition 6. Let U be a universal set and A be a subset of U. The complement of A,
denoted A, is
{x : x E U andx ý A}
Sometimes, to emphasize that U is a universal set, A is also called the absolute difference.
With this definition, we can restate Definition 5 as A - B = A nf B. Some important
identities concern complements, especially how they interact with other set-theoretic operations.
Theorem 7. Let U be a universal set and A and B be subsets of U. Then:
(a) A = A. (A is the complement of A.)
(b) A C B if and only if B C A.
(c) A = B if and only if A = B.
(What the proof entails.) Part (a) tells us that the complement only produces something
new the first time it is applied. Part (b) says that if A is a subset of B, then set inclusion
goes the other way for the complements; that is, the complement of B is contained in the
complement of A. In part (c), we prove that if two sets are equal, then their complements
are equal.