Set Theory

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Notations: ∈, {.. | ..} Let S be a set (a collection of elements, with no duplicates) —  a ∈ S means that a is an element of S

Example: 1 ∈ {1,2,3}, 3 ∈ {1,2,3} —  a ∉ S means that a is not an element of S

Example: 4 ∉ {1,2,3} —  A = {x ∈ S | P(x)} is the set of all elements x of S such that P(x)

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Outline —  Subsets —  Venn Diagrams —  Set Operations (Union, Intersection, …. ) —  Empty Set, Partitions, Power Set, Cartesian Product,

Cardinality. —  Set Properties and Identities —  Proofs of Set Properties. —  Proofs involving Empty Sets

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Subsets —  Definition:

A ⊆ B ⇔ ∀x, if x ∈ A then x ∈ B —  A ⊈ B ⇔ ∃x such that x ∈ A and x ∉ B —  A is a proper subset of B (A⊂B) ⇔ (1) A⊆B AND (2) there is at least one element in B that is not in A Examples: {1} ⊆ {1} {1} ⊆ {1, {1}} {1} ⊂ {1, 2} {1} ⊂ {1, {1}} 4

Proving Subset To prove that X ⊆ Y: 1.  Pick an element x. 2.  Assume, x is in X. 3.  Show that x is also in Y.

The above is called the “element argument”.

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Subset Proof Example A = {m ∈ Z|m = 6r + 12 for some r ∈ Z} B = {n ∈ Z | n = 3s for some s ∈ Z} Pick x. Suppose x is in A. We must show that x ∈ B.

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Subset: Disproof Example Prove B ⊈ A A = {m ∈ Z|m = 6r + 12 for some r ∈ Z} B = {n ∈ Z | n = 3s for some s ∈ Z} We must find an element of B that is not an element of A. Try 3.

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Set Equality: Proof A = B iff every element of A is in B and every element of B is in A. Thus, A = B ⇔ A ⊆ B and B ⊆ A Example: A = {m ∈ Z | m = 2a for some integer a} B = {n ∈ Z | n = 2b − 2 for some integer b}

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Outline —  Subsets —  Venn Diagrams —  Set Operations (Union, Intersection, …. ) —  Empty Set, Partitions, Power Set, Cartesian Product,

Cardinality. —  Set Properties and Identities —  Proofs of Set Properties. —  Proofs involving Empty Sets

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Venn Diagrams — A ⊆ B

— A ⊈ B

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Set of Integers, Rationals, Reals —  Z, Q, and R denote the sets of integers, rational numbers,

and real numbers —  Z ⊆ Q. —  Z is a proper subset of Q —  Q ⊆ R —  Q is a proper subset of R

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Operations on Sets Let A and B be subsets of a universal set U. 1. The union of A and B: A ∪ B is the set of all elements that are in at least one of A or B: A ∪ B = {x ∈ U | x ∈ A or x ∈ B} 2. The intersection of A and B: A ∩ B is the set of all elements that are common to both A and B. A ∩ B = {x ∈ U | x ∈ A and x ∈ B} 3. The difference of B minus A (relative complement of A in B): B −A (or B\A) is the set of all elements that are in B and not A. B − A = {x ∈ U | x ∈ B and x ∉ A} 4. The complement of A: Ac is the set of all elements in U that are not in A. Ac = {x ∈ U | x ∉ A} 12

Example Let U = {a, b, c, d, e, f, g}, A = {a, c, e, g} and B = {d, e, f, g}. A ∪ B = {a, c, d, e, f, g} A ∩ B = {e, g} B − A = {d, f } Ac = {b, d, f } 13

Example: Sets of real numbers —  Given real numbers a and b with a ≤ b: —  (a, b) = {x ∈ R | a < x < b} —  (a, b] = {x ∈ R | a < x ≤ b} —  [a, b) = {x ∈ R | a ≤ x < b} —  [a, b] = {x ∈ R | a ≤ x ≤ b} —  The symbols ∞ and −∞ are used to indicate intervals that are

unbounded either on the right or on the left: —  (a,∞)={x ∈ R | a < x} —  [a,∞) ={x ∈ R | a ≤ x} —  (−∞, b)={x ∈ R | x < b} —  (−∞, b]={x ∈ R | x ≤ b} 14

Sets of real numbers A = (−1, 0] = {x ∈ R|−1 < x ≤ 0} B = [0, 1) = {x ∈ R| 0 ≤ x < 1} A ∪ B = (−1, 1) A ∩ B = {0}. B = (0, 1) Ac = (−∞, −1] ∪ (0, ∞)

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Outline —  Subsets —  Venn Diagrams —  Set Operations (Union, Intersection, …. ) —  Empty Set, Partitions, Power Set, Cartesian Product,

Cardinality. —  Set Properties and Identities —  Proofs of Set Properties. —  Proofs involving Empty Sets

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The Empty Set: ∅ or { }

∅ = {} a set that has no elements Examples: {1,2} ∩ {3,4}= ∅ {x ∈ R| 3 < x < 2} = ∅

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Partitions of Sets —  A and B are disjoint if they have no elements in common,

i.e., A ∩ B = ∅ —  Sets A1, A2, A3,... are mutually disjoint if all pairs of sets Ai, Aj (i ≠ j) are disjoint, i.e., ∀ i,j, if i ≠ j then Ai∩ Aj = ∅ —  A collection of non-empty sets{A1,A2, A3,...} is a partition of a set A ó 1. A = Ai 2. A1,A2, A3,... are mutually disjoint

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Partitions of Sets Example 1 —  A = {1, 2, 3, 4, 5, 6}

A1 = {1, 2}, A2= {3, 4}, A3 = {5, 6}

Example 2 T1 = {n ∈ Z| n = 3k, for some integer k} T2 = {n ∈ Z| n = 3k + 1, for some integer k} T3 = {n ∈ Z| n = 3k + 2, for some integer k} {T1,T2, T3}is a partition of Z 19

Power Set Given a set A, the power set of A, P(A), is the set of all subsets of A Examples: P({x, y}) = {∅, {x}, {y}, {x, y}} P(∅) = {∅} P({∅}) = {∅, {∅}} 20

Cartesian Product —  An ordered n-tuple (x1,x2,...,xn) is an ordered list of elements

x1,x2,...,xn. —  Equality: Two ordered n-tuples (x1,x2,...,xn) and (y1,y2,...,yn) are equal if each of their corresponding elements are equal. (x1,x2,...,xn)=(y1,y2,...,yn) ó x1=y1and x2=y2 and ... xn=yn —  The cartesian product of A1,A2,...,An: A1×A2×... ×An={(a1, a2,..., an) | a1∈A1, a2∈A2,..., an∈An} Example: A={1,2}, B={3,4} A×B ={(1,3), (1,4), (2,3), (2,4)}

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Cartesian Product: Example A = {x, y}, B = {1, 2, 3}, and C = {a, b} A × B × C?

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Cardinality of a set — The cardinality of a set A: N(A) or|A| is a

measure of the "number of elements of the set" — Example: |{2, 4, 6}| = 3 — For any sets A and B, |A ∪ B| + |A ∩ B| = |A|+|B| — If A and B are disjoint sets, then

|A ∪ B| = |A|+|B| 23

The Size of the Power Set For all n ≥ 0, X has n elements à P(X) has 2n elements. Proof (by mathematical induction):

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Outline —  Subsets —  Venn Diagrams —  Set Operations (Union, Intersection, …. ) —  Empty Set, Partitions, Power Set, Cartesian Product,

Cardinality. —  Set Properties and Identities —  Proofs of Set Properties. —  Proofs involving Empty Sets

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Set Properties —  Inclusion of Intersection:

A ∩ B ⊆A and A ∩ B ⊆ B —  Inclusion in Union: A ⊆ A ∪ B and B ⊆ A ∪ B —  Transitive Property of Subsets: A ⊆ B and B ⊆ C à A ⊆ C —  x ∈ A ∪ B ⇔ x ∈ A or x ∈ B —  x ∈ A ∩ B ⇔ x ∈ A and x ∈ B —  x ∈ B − A ⇔ x ∈ B and x ∉ A —  x ∈ Ac ⇔ x ∉ A —  (x, y) ∈ A × B ⇔ x ∈ A and y ∈ B

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Set Identities —  Commutative Laws: A∪B = B∪A and A∩B = B∩A —  Associative Laws: (A∪B)∪C=A∪(B∪C) and (A∩B)∩C=A∩(B∩C) —  Distributive Laws: A∪(B∩C)=(A∪B)∩(A∪C), —  —  —  —  —  —  —  —  — 

A∩(B∪C)=(A∩B)∪(A∩C) Identity Laws: A∪∅ = A and A∩U = A Complement Laws: A∪Ac = U and A∩Ac = ∅ Double Complement Law: (Ac)c = A Idempotent Laws: A∪A = A and A∩A = A Universal Bound Laws: A ∪ U = U and A∩∅ = ∅ De Morgan’s Laws: (A ∪ B)c = Ac∩Bc and (A∩B)c = Ac ∪ Bc Absorption Laws: A ∪ (A ∩ B) = A and A ∩ (A ∪ B) = A Complements of U and ∅: Uc = ∅ and ∅c = U Set Difference Law: A − B = A ∩ Bc

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Outline —  Subsets —  Venn Diagrams —  Set Operations (Union, Intersection, …. ) —  Empty Set, Partitions, Power Set, Cartesian Product,

Cardinality. —  Set Properties and Identities —  Proofs of Set Properties. —  Proofs involving Empty Sets

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Proof of Inclusion Property A ∩ B ⊆ A. The statement to be proved is universal: ∀ sets A and B, A∩B ⊆ A Proof: ?

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Disproving a Set Property: Counterexample Example: For all sets A,B, and C, (A−B)∪(B−C) =

A−C ?

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Proof of a Set Identity For all sets A, B, and C, A∪(B∩C)=(A∪B)∩(A∪C) Proof?

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Proof of De Morgan’s Law For all sets A and B: (A∪B)c = Ac∩Bc Proof?

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Algebraic Proof of Set Identity – 1 Example: For all sets A, B, and C,(A∪B)−C=(A−C)∪(B−C). Algebraic proof: (A ∪ B) − C = (A ∪ B) ∩ Cc by the set difference law = Cc ∩ (A ∪ B) by the commutative law for ∩ = (Cc ∩ A) ∪ (Cc ∩ B) by the distributive law = (A ∩ Cc) ∪ (B ∩ Cc) by the commutative law for ∩ = (A − C) ∪ (B − C) by the set difference law.

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Algebraic Proof of Set Identity – 2

Example: for all sets A and B, A − (A ∩ B) = A − B. A − (A ∩ B) = A ∩ (A ∩ B)c by the set difference law = A ∩ (Ac ∪ Bc) by De Morgan’s laws = (A ∩ Ac) ∪ (A ∩ Bc) by the distributive law = ∅∪(A ∩ Bc) by the complement law = (A ∩ Bc) ∪ ∅ by the commutative law for ∪ = A ∩ Bc by the identity law for ∪ =A − B by the set difference law. 35

Outline —  Subsets —  Venn Diagrams —  Set Operations (Union, Intersection, …. ) —  Empty Set, Partitions, Power Set, Cartesian Product,

Cardinality —  Set Properties and Identities —  Proofs of Set Properties —  Proofs involving Empty Sets

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Proofs involving Empty Sets Prove: 1.  A set with no elements is a subset of every set. 2.  Uniqueness of the empty set. Hint: Use contradiction.

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Proving a set is empty To prove X = ∅, use contradiction. Example 1: For any set A, A ∩∅ = ∅. Example 2: For all sets A, B, and C, if A ⊆ B and B ⊆ Cc, then A ∩ C = ∅.

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Intersection and Union with a Subset Prove: For any sets A and B, if A ⊆ B, then A∩B=A and A∪B=B Proof: A∩B = A ó (1) A ∩ B ⊆ A and (2) A ⊆ A ∩ B A∪B = B ó (3) A ∪ B ⊆ B and (4) B ⊆ A ∪ B So, prove each of the above four properties.

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