The document discusses set theory concepts including: - A set is a collection of elements with no order or repetition. - Two sets are equal if they contain the same elements. - A subset is defined as all elements of one set also being elements of another set. - Examples of common sets like natural numbers, integers, and real numbers are provided. The document provides formal definitions and examples of sets, subsets, equality of sets, and proper subsets.

1. January 29, Applied Discrete Mathematics 1
Set TheorySet Theory
..

2. January 29, Applied Discrete Mathematics 2
Set TheorySet Theory
• Set: Collection of objects (“elements”)Set: Collection of objects (“elements”)
• aa∈∈AA “a is an element of A”“a is an element of A”
“a is a member of A”“a is a member of A”
• aa∉∉AA “a is not an element of A”“a is not an element of A”
• A = {aA = {a11, a, a22, …, a, …, ann}} “A contains…”“A contains…”
• Order of elements is meaninglessOrder of elements is meaningless
• It does not matter how often the same elementIt does not matter how often the same element
is listed.is listed.

3. January 29, Applied Discrete Mathematics 3
Set EqualitySet Equality
Sets A and B are equal if and only if they containSets A and B are equal if and only if they contain
exactly the same elements.exactly the same elements.
Examples:Examples:
• A = {9, 2, 7, -3}, B = {7, 9, -3, 2} :A = {9, 2, 7, -3}, B = {7, 9, -3, 2} : A = BA = B
• A = {dog, cat, horse},A = {dog, cat, horse},
B = {cat, horse, squirrel, dog} :B = {cat, horse, squirrel, dog} : AA ≠≠ BB
• A = {dog, cat, horse},A = {dog, cat, horse},
B = {cat, horse, dog, dog} :B = {cat, horse, dog, dog} : A = BA = B

4. January 29, Applied Discrete Mathematics 4
Examples for SetsExamples for Sets
““Standard” Sets:Standard” Sets:
• Natural numbersNatural numbers NN = {0, 1, 2, 3, …}= {0, 1, 2, 3, …}
• IntegersIntegers ZZ = {…, -2, -1, 0, 1, 2, …}= {…, -2, -1, 0, 1, 2, …}
• Positive IntegersPositive Integers ZZ++
= {1, 2, 3, 4, …}= {1, 2, 3, 4, …}
• Real NumbersReal Numbers RR = {47.3, -12,= {47.3, -12, ππ, …}, …}
• Rational NumbersRational Numbers QQ = {1.5, 2.6, -3.8, 15, …}= {1.5, 2.6, -3.8, 15, …}
(correct definition will follow)(correct definition will follow)

5. January 29, Applied Discrete Mathematics 5
Examples for SetsExamples for Sets
• A =A = ∅∅ “empty set/null set”“empty set/null set”
• A = {z}A = {z} Note: zNote: z∈∈A, but zA, but z ≠≠ {z}{z}
• A = {{b, c}, {c, x, d}}A = {{b, c}, {c, x, d}}
• A = {{x, y}}A = {{x, y}}
Note: {x, y}Note: {x, y} ∈∈A, but {x, y}A, but {x, y} ≠≠ {{x, y}}{{x, y}}
• A = {x | P(x)}A = {x | P(x)}
“set of all x such that P(x)”“set of all x such that P(x)”
• A = {x | xA = {x | x∈∈NN ∧∧ x > 7} = {8, 9, 10, …}x > 7} = {8, 9, 10, …}
“set builder notation”“set builder notation”

6. January 29, Applied Discrete Mathematics 6
Examples for SetsExamples for Sets
We are now able to define the set of rationalWe are now able to define the set of rational
numbers Q:numbers Q:
QQ = {a/b | a= {a/b | a∈∈ZZ ∧∧ bb∈∈ZZ++
}}
oror
QQ = {a/b | a= {a/b | a∈∈ZZ ∧∧ bb∈∈ZZ ∧∧ bb≠≠0}0}
And how about the set of real numbers R?And how about the set of real numbers R?
RR = {r | r is a real number}= {r | r is a real number}
That is the best we can do.That is the best we can do.

7. January 29, Applied Discrete Mathematics 7
SubsetsSubsets
AA ⊆⊆ BB “A is a subset of B”“A is a subset of B”
AA ⊆⊆ B if and only if every element of A is alsoB if and only if every element of A is also
an element of B.an element of B.
We can completely formalize this:We can completely formalize this:
AA ⊆⊆ BB ⇔⇔ ∀∀x (xx (x∈∈AA →→ xx∈∈B)B)
Examples:Examples:
A = {3, 9}, B = {5, 9, 1, 3}, AA = {3, 9}, B = {5, 9, 1, 3}, A ⊆⊆ B ?B ? truetrue
A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, AA = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A ⊆⊆ B ?B ?
falsefalse
truetrue
A = {1, 2, 3}, B = {2, 3, 4}, AA = {1, 2, 3}, B = {2, 3, 4}, A ⊆⊆ B ?B ?

8. January 29, Applied Discrete Mathematics 8
SubsetsSubsets
Useful rules:Useful rules:
• A = BA = B ⇔⇔ (A(A ⊆⊆ B)B) ∧∧ (B(B ⊆⊆ A)A)
• (A(A ⊆⊆ B)B) ∧∧ (B(B ⊆⊆ C)C) ⇒⇒ AA ⊆⊆ CC (see Venn Diagram)(see Venn Diagram)
UU
AA
BB
CC

9. January 29, Applied Discrete Mathematics 9
SubsetsSubsets
Useful rules:Useful rules:
∀ ∅∅ ⊆⊆ A for any set AA for any set A
• AA ⊆⊆ A for any set AA for any set A
Proper subsets:Proper subsets:
AA ⊂⊂ BB “A is a proper subset of B”“A is a proper subset of B”
AA ⊂⊂ BB ⇔⇔ ∀∀x (xx (x∈∈AA →→ xx∈∈B)B) ∧∧ ∃∃x (xx (x∈∈BB ∧∧ xx∉∉A)A)
oror
AA ⊂⊂ BB ⇔⇔ ∀∀x (xx (x∈∈AA →→ xx∈∈B)B) ∧∧ ¬∀¬∀x (xx (x∈∈BB →→ xx∈∈A)A)

10. January 29, Applied Discrete Mathematics 9
SubsetsSubsets
Useful rules:Useful rules:
∀ ∅∅ ⊆⊆ A for any set AA for any set A
• AA ⊆⊆ A for any set AA for any set A
Proper subsets:Proper subsets:
AA ⊂⊂ BB “A is a proper subset of B”“A is a proper subset of B”
AA ⊂⊂ BB ⇔⇔ ∀∀x (xx (x∈∈AA →→ xx∈∈B)B) ∧∧ ∃∃x (xx (x∈∈BB ∧∧ xx∉∉A)A)
oror
AA ⊂⊂ BB ⇔⇔ ∀∀x (xx (x∈∈AA →→ xx∈∈B)B) ∧∧ ¬∀¬∀x (xx (x∈∈BB →→ xx∈∈A)A)