Section 4.1 Sets
249
c. A c (B d A) = A
d. (A d B′)′ c B = A′ c B
e. (A d B) c (A d B′) = A
f. [A d (B c C )]′ = A′ c (B′ d C′)
83. A, B, and C are subsets of a set S. Prove the following set identities using the basic set identities listed in
this section. Give a reason for each step. State the dual of each of these identities.
a. (A c B) d (A c B′) = A
b. ([(A d C ) d B] c [(A d C ) d B′]) c (A d C )′ = S
c. (A c C ) d [(A d B) c (C′ d B)] = A d B
84. A is a subset of a set S. Prove the following set identities:
a. A c A = A
d. A c S = S
b. A d A = A
e. (A′)′ = A
c. A d [ = [
85. A, B, and C are subsets of a set S. Prove the following set identities by using previously proved identities,
including those in Exercises 82–84. Give a reason for each step.
a. A d (B c A′) = B d A
b. (A c B) − C = (A − C ) c ( B − C )
c. (A − B) − C = (A − C ) − B
86. A, B, and C are subsets of a set S. Prove the following set identities by using previously proved identities,
including those in Exercises 82–84. Give a reason for each step.
a. [(A′ c B′) d A′ ]′ = A
b. (A − B) − C = (A − C ) − (B − C )
c. A − (A − B) = A d B
d. (A c B) − (A d B) = (A − B) c (B − A)
87. The operation of set union can be defined as an n-ary operation for any integer n ≥ 2.
a. Give a definition similar to that for the union of two sets for A 1 c A 2 c c c A n .
b. Give a recursive definition for A 1 c A 2 c c c A n .
88. Using the recursive definition of set union from Exercise 87(b), prove the generalized associative property
of set union, which is that for any n with n ≥ 3 and any p with 1 ≤ p ≤ n − 1,
(A 1 c A 2 c c c A p ) c (A p+1 c A p+2 c c c A n ) = A 1 c A 2 c c c A n
89. The operation of set intersection can be defined as an n-ary operation for any integer n ≥ 2.
a. Give a definition similar to that for the intersection of two sets for A 1 d A 2 d c d A n .
b. Give a recursive definition for A 1 d A 2 d c d A n .
90. Using the recursive definition of set intersection from Exercise 89(b), prove the generalized associative
property of set intersection, which is that for any n with n ≥ 3 and any p with 1 ≤ p ≤ n − 1,
(A 1 d A 2 d c d A p ) d (A p+1 d A p+2 d c d A n ) = A 1 d A 2 d c d A n
249
c. A c (B d A) = A
d. (A d B′)′ c B = A′ c B
e. (A d B) c (A d B′) = A
f. [A d (B c C )]′ = A′ c (B′ d C′)
83. A, B, and C are subsets of a set S. Prove the following set identities using the basic set identities listed in
this section. Give a reason for each step. State the dual of each of these identities.
a. (A c B) d (A c B′) = A
b. ([(A d C ) d B] c [(A d C ) d B′]) c (A d C )′ = S
c. (A c C ) d [(A d B) c (C′ d B)] = A d B
84. A is a subset of a set S. Prove the following set identities:
a. A c A = A
d. A c S = S
b. A d A = A
e. (A′)′ = A
c. A d [ = [
85. A, B, and C are subsets of a set S. Prove the following set identities by using previously proved identities,
including those in Exercises 82–84. Give a reason for each step.
a. A d (B c A′) = B d A
b. (A c B) − C = (A − C ) c ( B − C )
c. (A − B) − C = (A − C ) − B
86. A, B, and C are subsets of a set S. Prove the following set identities by using previously proved identities,
including those in Exercises 82–84. Give a reason for each step.
a. [(A′ c B′) d A′ ]′ = A
b. (A − B) − C = (A − C ) − (B − C )
c. A − (A − B) = A d B
d. (A c B) − (A d B) = (A − B) c (B − A)
87. The operation of set union can be defined as an n-ary operation for any integer n ≥ 2.
a. Give a definition similar to that for the union of two sets for A 1 c A 2 c c c A n .
b. Give a recursive definition for A 1 c A 2 c c c A n .
88. Using the recursive definition of set union from Exercise 87(b), prove the generalized associative property
of set union, which is that for any n with n ≥ 3 and any p with 1 ≤ p ≤ n − 1,
(A 1 c A 2 c c c A p ) c (A p+1 c A p+2 c c c A n ) = A 1 c A 2 c c c A n
89. The operation of set intersection can be defined as an n-ary operation for any integer n ≥ 2.
a. Give a definition similar to that for the intersection of two sets for A 1 d A 2 d c d A n .
b. Give a recursive definition for A 1 d A 2 d c d A n .
90. Using the recursive definition of set intersection from Exercise 89(b), prove the generalized associative
property of set intersection, which is that for any n with n ≥ 3 and any p with 1 ≤ p ≤ n − 1,
(A 1 d A 2 d c d A p ) d (A p+1 d A p+2 d c d A n ) = A 1 d A 2 d c d A n