Foundations of Discrete Mathematics

Foundations of Discrete Mathematics Chapter 2 By Dr. Dalia M. Gil, Ph.D.

Sets • Set: A collection of things called elements or members. • The set of natural numbers N consists of the numbers 1, 2,... Their members are all positive.

Sets • The set of integers Z consists of the natural numbers (1, 2, …), their negatives (…, -3, -2, …, 2, 3, …, and 0. • Zero (0) is an integer, but not a natural number.

Ways to Describe Sets • {egg1, egg2} • {x} • N = {1, 2, 3, …} • Z = {…, -3, -2, -1, 0, 1, 2, 3, …}  This set has two elements  This set has one element  The set of natural numbers  The set of integer numbers

Describing a Set with a Builder Notation { x|x has certain properties } such that We read: “The set of x such that x has certain properties.”

Describing a Set with a Builder Notation { some expression | the expression has certain properties } Example: the set of odd natural numbers. {n | n is an odd integer, n > 0} such that

Describing a Set with a Builder Notation Example: the set of odd natural numbers. {2k – 1 | k = 1, 2, 3, …} or {2k – 1 | k  N} K belongs to N

Describing a Set with a Builder Notation • The symboldenoting set membership m  Z  m is an integer • negates the meaning of 0  N

Describing a Set with a Builder Notation • The set of rational numbers Q Q = {m/n| m, n  Z, n ≠ 0} • The members of Q are fractions, which are ratios of integers with nonzero denominators. • Examples ¾, -2/17, 5(=5/1)

Describing a Set with a Builder Notation • The set of irrational numbers. • The members of irrational set cannot be written in the form m/n with m and n both integers. • The decimal expansions of the irrational numbers neither terminate or repeat. • Examples √2, 3√17, e, , ln 5

Describing a Set with a Builder Notation • The set of complex numbers C. • The members of complex set have the form a + bi where a and b are real numbers, i2 = -1 C = {a + bi | a, b  R, i2 = -1}

Describing a Set with a Builder Notation • A set can be an element of another set {{a, b}, c}  is a set with two elements, {a, b} and c.

Equality of Set • Sets A and B are equal, and we write A = B, if and only if A and B contain the same elements or neither set contains any element. • {1, 2, 1} = {1, 2} = {2, 1} • {1/2, 2/4, -3/-6}, /2} = {1/2} • {t|t = r – s, r, s {0, 1, 2}} = {-2, -1, 0, 1, 2}

The Empty Set • The empty set is a set that contains no elements. • P = {n  N | 5n = 2} • S = {n  N | n2 + 1 = 0} • The set small of people less than 1 millimeter. These sets are all equal since none of them contains any elements.

Subsets • A set A is a subset of a set B (A  B), if and only if every element of A is an element of B. • If A  B but A ≠ B, then A is called a proper subset of B and we write A  B ≠

Subsets • A  B  A is contained in B  A is a subset of B • B  A  B is superset of A

Examples of Subsets • {a, b}  {a, b, c} {a,b} is a subset of {a,b,c} • {a, b}  {a, b, c} ≠ {a,b} is a proper subset of {a,b,c}

Examples of Subsets • {a, b}  {a, b, {a, b}} {a,b} is a subset of {a,b,{a,b}} {a,b} is an element of {a,b,{a,b}} • {a, b}  {a, b, {a, b}} {a,b} belongs to {a,b,{a,b}}

Examples of Subsets N  Z  Q  R  C ≠ ≠ ≠ ≠ • The set of natural numbers is a proper subset of the set of integer numbers. • The set of integer numbers is a proper subset of the set of rational numbers.

Examples of Subsets N  Z  Q  R  C ≠ ≠ ≠ ≠ • The set of rational numbers is a proper subset of the set of real numbers. • The set of real numbers is a proper subset of the set of complex numbers.

Subsets. Proposition 1 • For any set A, A  A and   A Proof • If a  A, then a  A, so A  A • If   A is false, then there must exist some x   such that x  A. This an absurdity since there is no x  

Subsets. Proposition 2 • If A and B are sets, then A = B if and only if A  B and B  A Proof () If A = B, then A is a subset of B and B is a subset of A. () If A is a subset of B and B is a subset of A, then A = B.

Subsets. Proposition 2 • a  b membership a  b, a is an element of the set b. • a  b subset a  b, a is a set each of whose elements is also in the set b.

The Power Set • The power set of a set A, denoted P(A), is the set of all subsets of A: P(A) = {B | B  A}

Examples of The Power Set • If A = {a}, then P(A) = {, {a}} • If A = {a, b}, then P(A) = {, {a}, {b}, {a, b}} • P ({a, b, c}) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}

Union and Intersection • The union of sets A and B, A  B, is the set of elements in A or in B (or in both). • The intersection of sets A and B, A  B, is the set of elements that belongs to both A and B.

Examples: Union and Intersection • If A = {a, b, c} and b = {a, x, y, b} • A  B = { a, b, c, x, y} • A  B = {a, b} • A  {} = {a, b, c, } • B  {} = 

Examples: Union and Intersection • For any set A, A   = A and A   = 

Union and Intersection • The union and intersection of sets are associative operations. • (A1  A2)  A3 = A1  (A2  A3) • For any three sets A1, A2, A3 , the expression A1  A2  A3 is unambiguous.

Union and Intersection • The union of n sets A1  A2  A3 … An is written n  Ai i=1 • Represents the set of elements that belong to one or more of the sets Ai

Union and Intersection • The intersection the sets A1, A2, … An is written n  Ai i=1 • Represents the set of elements which belong to all of the sets Ai

Union and Intersection • A = {1, 2, 3, 4} • B = {3, 4, 5, 6} • C = {2, 3, 5, 7} • B  C= {2, 3, 4, 5, 6, 7} • A  (B  C)= {2, 3, 4}

Union and Intersection • A = {1, 2, 3, 4} • B = {3, 4, 5, 6} • C = {2, 3, 5, 7} • A  B = {3, 4} • (A  B)  C= {2, 3, 4, 5, 7}

Union and Intersection A  (B  C)= {2, 3, 4} (A  B)  C= {2, 3, 4, 5, 7} A  (B  C) ≠ (A  B)  C

Union and Intersection A  (B  C)= {2, 3, 4} (A  B) = {3, 4} (A  C)= {2, 3} (A  B)  (A  C)= {2, 3, 4} A  (B  C) = (A  B)  (A  C)

Union and Intersection Let A, B, and C be sets. Verify A  (B  C) = (A  B)  (A  C) Solution using proposition 2: If A and B are sets, then A = B if and only if A  B and B  A

Union and Intersection To show A, B, and C be sets. Verify A  (B  C)  (A  B)  (A  C) Let x A  (B  C) Then x is in A and also in B  C, Since x B  C, either x  B or x  C. This suggests two cases

Union and Intersection Case 1: x  B In this case, is in A as well as in B, so it’s in A  B Case 2: x  C In this case, is in A as well as in B, so it’s in A  C

Union and Intersection We have shown that either x  A  Borx  A  C By definition of union, x  (A  B)  (A  C)

Union and Intersection We must show thatA  (B  C)  (A  B)  (A  C) Let x  (A  B)  (A  C) Then either x  (A  B) or x  (A  C) Thus, x is in both A and B or in both A and C. In either case x  A. Also x is in either B or C; thusx  B  C

Union and Intersection We must show thatA  (B  C)  (A  B)  (A  C) Let x  (A  B)  (A  C) Then either x  (A  B) or x  (A  C) So x is in both A and in B  C ; that is x  A  ( B  C).

Set Difference • The set difference of sets A and B (A\ B), is the set of those elements of A that are not in B. • The complement of a set A is the set Ac = U \ A, where U is some universal set made clear by the context.

Examples: Set Difference • {a, b, c} \ {a, b} = {c} • {a, b, c} \ {a, x} = {b, c} • {a, b, } \  = {a, b}

Examples: Set Difference • {a, b, } \ {} = {a, b, } • If A is the set {Monday, Tuesday, Wednesday, Thursday, Friday}, so • Ac = {Saturday, Sunday}

Examples: Set Difference • A \ B = A  Bc and (Ac)c= A Example (suppose U = Z): If A = {x  Z | x2 > 0}, then Ac ={0} (Ac)c = {0}c = {x  Z | x ≠ 0} = A

Interval Notation • If a and b are real numbers with a < b, then • [a, b] = {x  R | a ≤ x ≤ b} closed • (a, b) = {x  R | a < x < b} open

Interval Notation • If a and b are real numbers with a < b, then • (a, b] = {x  R | a < x ≤ b} half open • [a, b) = {x  R | a ≤ x < b} half open

The Laws of De Morgan • (A  B)c = Ac  Bc • (A  B)c = Ac Bc

Prove that (A  B)c = Ac  Bcfor any set A, B, and C. • Let A be the statement “x  A” and B be the statement “x  B” x  (A  B)c  ¬(x  A  B)

Prove that (A  B)c = Ac  Bc for any set A, B, and C. x  (A  B)c  ¬(x  A  B)  ¬(A or B) Definition of union  ¬A and ¬B Rule for negating “or”  x  Ac and x  Bc  x  Ac  Bc Definition of intersection

Foundations of Discrete Mathematics