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Transcript of Discrete Structures
6.2 Properties of Sets 1
Discrete Structures
Chapter 6: Set Theory
6.2 Properties of Sets
…only the last line is a genuine theorem here – everything else is in the fantasy.
– Douglas Hofstadter, 1945 – present Gödel, Escher, Bach: an Eternal Golden Braid, 1979
Erickson
6.2 Properties of Sets 2
Theorem 6.2.1 – Some Subset Relations
Erickson
6.2 Properties of Sets 3
Procedural Versions of Set Definitions
Erickson
6.2 Properties of Sets 4
Theorem 6.2.2 – Set Identities
Let all sets referred to below be subsets of a universal set U.
1. Commutative Laws: For all sets A and B,
a. A B = B A
b. A B = B A
2. Associative Laws: For all sets A, B, and C,
a. (A B) C = A (B C)
b. (A B) C = A (B C)
Erickson
6.2 Properties of Sets 5
Theorem 6.2.2 – Set Identities
Let all sets referred to below be subsets of a universal set U.
3. Distributive Laws: For all sets A, B, and C,
a. A (B C) = (A B) (A C)
b. A (B C) = (A B) (A C)
4. Identity Laws: For all sets A,
a. A = A
b. A U = A
Erickson
6.2 Properties of Sets 6
Theorem 6.2.2 – Set Identities
Let all sets referred to below be subsets of a universal set U.
5. Complement Laws:
a. A Ac = U
b. A Ac = 6. Double Complement Laws: For all sets A,
(Ac)c = A
Erickson
6.2 Properties of Sets 7
Theorem 6.2.2 – Set Identities
Let all sets referred to below be subsets of a universal set U.
7. Idempotent Laws: For all sets A,
a. A A = A
b. A A = A
8. Universal Bound Laws: For all sets A,
a. A U = U
b. A =
Erickson
6.2 Properties of Sets 8
Theorem 6.2.2 – Set Identities
Let all sets referred to below be subsets of a universal set U.
9. De Morgan’s Laws: For all sets A, and B,
a. (A B)c = Ac Bc
b. (A B)c = Ac Bc
10. Absorption Laws: For all sets A and B,
a. A (A B) = A
b. A (A B) = A
Erickson
6.2 Properties of Sets 9
Theorem 6.2.2 – Set Identities
Let all sets referred to below be subsets of a universal set U.
11. Complements of U and :
a. Uc = b. c = U
12. Set Difference Laws: For all sets A and B,
A – B = A Bc
Erickson
6.2 Properties of Sets 10
Proving Sets are Equal
• The Basic Method for Proving Sets are Equal
Let sets X and Y be given. To prove that X = Y:
1. Prove that X Y.
2. Prove that Y X.
Erickson
6.2 Properties of Sets 11
Theorem 6.2.3 – Intersection and Union with a Subset
For any sets A and B, if A B, then
a. A B = A
b. A B = B
Erickson
6.2 Properties of Sets 12
Theorem 6.2.4
• Theorem 6.2.4 – A Set with no Elements is a Subset of Every Set
If E is a set with no elements and A is any set, then E A.
Erickson
6.2 Properties of Sets 13
Corollary 6.2.5
• Corollary 6.2.5 – Uniqueness of the Empty Set
There is only one set with no elements.
Erickson
6.2 Properties of Sets 14
Proving a Set Equals the Empty Set
• Element Method to prove that a Set Equals the Empty Set
To prove that a set X is equal to the empty set , prove that X has no elements.
To do this, suppose X has an element and derive a contradiction.
Erickson
6.2 Properties of Sets 15
Proposition 6.2.6
• For all sets A, B, and C, if A B and B Cc, then A C = .
Erickson
6.2 Properties of Sets 16
Examples – pg. 365
• Use an element argument to prove each of the following statements. Assume that all sets are the subsets of a universal set U.
7. For all sets and , .
9. For all sets , , and , .
11. For all sets and , .
16. For all sets , , and , if and then
.
c c cA B A B A B
A B C A B C B A B C
A B A A B A
A B C A B A C
A B C
Erickson