1. Discrete Structures Chapter 6: Set Theory
6.1 Set Theory: Definitions and the Element Method of Proof
The introduction of suitable abstractions is our only mental aid to organize and master complexity.
– E. W. Dijkstra, 1930 – 2002
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6.1 Set Theory - Definitions and the Element Method of Proof

2. 6.1 Set Theory - Definitions and the Element Method of Proof
Subsets
Let’s write what it means for a set A to be a subset of a set B as a formal universal conditional statement:
A  B  x, if x  A then x  B.
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6.1 Set Theory - Definitions and the Element Method of Proof

3. 6.1 Set Theory - Definitions and the Element Method of Proof
Subsets
The negation is existential
A  B  x, if x  A and x  B.
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6.1 Set Theory - Definitions and the Element Method of Proof

4. 6.1 Set Theory - Definitions and the Element Method of Proof
Subsets
A proper subset of a set is a subset that is not equal to its containing set.
A is a proper subset of B 
A  B, and
there is at least one element in B that is not in A.
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6.1 Set Theory - Definitions and the Element Method of Proof

5. 6.1 Set Theory - Definitions and the Element Method of Proof
Element Argument
Let sets X and Y be given. To prove that X  Y,
Suppose that x is a particular but arbitrarily chosen element of X,
Show that x is an element of Y.
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6.1 Set Theory - Definitions and the Element Method of Proof

6. 6.1 Set Theory - Definitions and the Element Method of Proof
Example – pg. 350 # 4
Let A = {n | n = 5r for some integer r} and B = {m | m = 20s for some integer s}.
Is A  B? Explain.
Is B  A? Explain.
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6.1 Set Theory - Definitions and the Element Method of Proof

7. 6.1 Set Theory - Definitions and the Element Method of Proof
Set Equality
Given sets A and B, A equals B, written A = B, iff every element of A is in B and every element of B is in A. Symbolically,
A = B  A  B and B  A
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6.1 Set Theory - Definitions and the Element Method of Proof

8. 6.1 Set Theory - Definitions and the Element Method of Proof
Operations on Sets
Let A and B be subsets of a universal set U.
1. The union of A and B denoted A  B, is the set of all elements that are in at least one of A or B.
Symbolically:
A  B = {x  U | x  A or x  B}
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6.1 Set Theory - Definitions and the Element Method of Proof

9. 6.1 Set Theory - Definitions and the Element Method of Proof
Operations on Sets
Let A and B be subsets of a universal set U.
2. The intersection of A and B denoted A  B, is the set of all elements that are common to both A or B.
Symbolically:
A  B = {x  U | x  A and x  B}
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6.1 Set Theory - Definitions and the Element Method of Proof

10. 6.1 Set Theory - Definitions and the Element Method of Proof
Operations on Sets
Let A and B be subsets of a universal set U.
3. The difference of B minus A (or relative complement of A in B) denoted B – A, is the set of all elements that are in B but not A.
Symbolically:
B – A = {x  U | x  B and x  A}
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6.1 Set Theory - Definitions and the Element Method of Proof

11. 6.1 Set Theory - Definitions and the Element Method of Proof
Operations on Sets
Let A and B be subsets of a universal set U.
4. The complement of A denoted Ac, is the set of all elements in U that are not A.
Symbolically:
Ac = {x  U | x  A}
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6.1 Set Theory - Definitions and the Element Method of Proof

12. 6.1 Set Theory - Definitions and the Element Method of Proof
Example – pg. 350 # 11
Let the universal set be the set R of all real numbers and let
A = {x  R | 0 < x  2}, B = {x  R | 1  x < 4}, and C = {x  R | 3  x < 9}. Find each of the following:
a. A  B b. A  B c. Ac
d. A  C e. A  C f. Bc
g. Ac  Bc h. Ac  Bc i. (A  B)c
j. (A  B)c
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6.1 Set Theory - Definitions and the Element Method of Proof

13. Unions and Intersections of an Indexed Collection of Sets
Given sets A0, A1, A2, … that are subsets of a universal set U and given a nonnegative integer n,
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6.1 Set Theory - Definitions and the Element Method of Proof

14. Definitions
Empty Set
A set with no elements is called the empty set (or null set) and denoted by the symbol .
Disjoint
Two sets are called disjoint iff they have no elements in common. Symbolically:
A and B are disjoint  A  B = 
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6.1 Set Theory - Definitions and the Element Method of Proof

15. 6.1 Set Theory - Definitions and the Element Method of Proof
Mutually Disjoint
Sets A1, A2, A3, … are mutually disjoint (or pairwise disjoint or nonoverlapping) iff no two sets Ai and Aj with distinct subscripts have any elements in common. More precisely, for all i, j = 1, 2, 3, …
Ai  Aj =  whenever i  j.
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6.1 Set Theory - Definitions and the Element Method of Proof

16. 6.1 Set Theory - Definitions and the Element Method of Proof
Example – pg. 305 # 23
Let for all positive integers i.
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6.1 Set Theory - Definitions and the Element Method of Proof

17. 6.1 Set Theory - Definitions and the Element Method of Proof
Partition
A finite or infinite collection of nonempty sets {A1, A2, A3, …} is a partition of a set A iff,
A is the union of all the Ai
The sets A1, A2, A3, …are mutually disjoint.
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6.1 Set Theory - Definitions and the Element Method of Proof

18. 6.1 Set Theory - Definitions and the Element Method of Proof
Example – pg. 351 # 27
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6.1 Set Theory - Definitions and the Element Method of Proof

19. 6.1 Set Theory - Definitions and the Element Method of Proof
Power Set
Given a set A, the power set of A is denoted (A), is the set of all subsets of A.
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6.1 Set Theory - Definitions and the Element Method of Proof

20. 6.1 Set Theory - Definitions and the Element Method of Proof
Example – pg. 351 # 31
Suppose A = {1, 2} and B = {2, 3}. Find each of the following:
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6.1 Set Theory - Definitions and the Element Method of Proof