M ACM 101
M ACM 101
Set Theory
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M ACM 101
Set Theory
Example
Example - Direct Proof
Prove that for all integers k and l, if k and l are both odd, then
their product kl is also odd.
Prove that if m is an even integer, then m + 7 is odd.
Set Theory
Example cont’d… - Contraposition
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M ACM 101
Set Theory
Example cont’d… - Contradiction
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M ACM 101
Set Theory
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M ACM 101
Set Theory
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Approach to Proofs
Example
If you are not given specific directions on how to do the
proof, try the following:
Prove that for all positive real numbers x and y, if the product
xy exceeds 25, then x > 5 or y > 5.
!
Try a direct proof
If you suspect the theorem is invalid, try to find a
counterexample
!
!
If that fails, try an indirect proof:
•
•
M ACM 101
Set Theory
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M ACM 101
Set Theory
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Chapter 3: Set Theory
Definitions
A set is a well-defined collection of objects. These objects
are called elements and are said to be members of the set.
If C, D are sets from a universe, we say that C is a subset of
D and write C ! D, or D " C if every element of C is an
element of D.
We use capital letters such as A, B, C, … to represent sets
and lowercase letters to represent elements.
For a set A we write x ! A if x is an element of A.
E.g. of sets
A = {1,3,5,7,9}
B = {x | x ! Z+, x is odd}
C = {1,3,5,7,9}
If, in addition, D contains an element that is not in C, then C
is called a proper subset of D, and this is denoted by C # D
or D $ C.
The size, or cardinality, of a finite set A is the number of
elements in the set, and is denoted | A |
E.g.
M ACM 101
Set Theory
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M ACM 101
Set Theory
Subsets
Set Equality
Note that for all sets C, D, C ! D if and only if
"x [x ! C # x ! D].
For a given universe, the sets C and D are said to be equal,
and we write C = D, when C ! D and D ! C.
E.g.
E.g.
M ACM 101
Set Theory
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M ACM 101
Set Theory
Theorem
Theorem - cont’d…
For a given universe containing the sets A, B, C:
(b) If A # B and B ! C, then A # C.
(a)
If A ! B and B ! C, then A ! C.
Proof:
Proof:
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M ACM 101
Set Theory
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M ACM 101
Set Theory
Theorem - cont’d…
Definitions
(c) If A ! B and B # C, then A # C.
The null set, or empty set, is the set containing no elements
and is denoted by % or {}.
(d) If A # B and B # C, then A # C.
E.g.
For the set A, the power set of A, denoted P(A), is the set of
all subsets of A.
E.g. C = {1,2,3,4}
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