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1 SETS

2 SET DEFINITION 1 Unordered collection of objects, called elements or members of the set. Said to contain its elements. We write a A to denote that a is an element of the set A. The notation a A denotes that a is not an element of the set A.

3 WAYS OF DESCRIBING SETS Roster Method Examples: set V of all vowels in the English alphabet can be written as V = {a, e, i, o, u}. The set O of odd positive integers less than 10 can be expressed by O = {1, 3, 5, 7, 9}. The set of positive integers less than 100 can be denoted by {1, 2, 3,..., 99}.

4 WAYS OF DESCRIBING SETS Set-Builder Notation Examples: O = {x x is an odd positive integer less than 10}, O = {x Z+ x is odd and x < 10}. Q+ = {x R x = p/q, for some positive integers p and q}.

5 OFTEN USED SETS N = {0, 1, 2, 3,...}, the set of natural numbers Z = {..., 2, 1, 0, 1, 2,...}, the set of integers Z+ = {1, 2, 3,...}, the set of positive integers

6 OFTEN USED SETS Q = {p/q p Z, q Z, and q = 0}, the set of rational numbers R, the set of real numbers R+, the set of positive real numbers C, the set of complex numbers.

7 Notation for intervals of Real Numbers When a and b are real numbers with a < b, we write: [a, b] = {x a x b} [a, b) = {x a x < b} Note : [a, b] is called the closed interval from a to b

8 Notation for intervals of Real Numbers When a and b are real numbers with a < b, we write: (a, b] = {x a < x b} (a, b) = {x a < x < b} Note : (a, b) is called the open interval from a to b.

9 EQUAL SETS DEFINITION 2 Two sets are equal if and only if they have the same elements. Therefore, if A and B are sets, then A and B are equal if and only if x(x A x B). We write A = B if A and B are equal sets. Example: sets {1, 3, 5} and {3, 5, 1} are equal

10 THE EMPTY SET Special set that has no elements Also called null set denoted by or { }

11 SINGLETON SET set with one element is called a singleton set. A common error is to confuse the empty set with the set { }, which is a singleton set. The single element of the set { } is the empty set itself!

12 VENN DIAGRAM Sets can be represented graphically using Venn diagrams, named after the English mathematician John Venn, who introduced their use in 1881.

13 VENN DIAGRAM In Venn diagrams the universal set U, which contains all the objects under consideration, is represented by a rectangle. Inside this rectangle, circles or other geometrical figures are used to represent sets. Sometimes points are used to represent the particular elements of the set.

14 VENN DIAGRAM Venn diagrams are often used to indicate the relationships between sets. Example: Venn diagram that represents V, the set of vowels in the English alphabet

15 SUBSETS DEFINITION 3 The set A is a subset of B if and only if every element of A is also an element of B. We use the notation A B to indicate that A is a subset of the set B.

16 EXAMPLES OF SUBSETS Set of all odd positive integers less than 10 is a subset of the set of all positive integers less than 10 Set of rational numbers is a subset of the set of real numbers, Set of all computer science majors at your school is a subset of the set of all students at your school

17 THEOREM 1 For every set S, (i) S and (ii) S S.

18 PROPER SUBSET When we wish to emphasize that a set A is a subset of a set B but that A = B, we write A B and say that A is a proper subset of B. For A B to be true, it must be the case that A B and there must exist an element x of B that is not an element of A. That is, A is a proper subset of B if and only if x(x A x B) x(x B x A)

19 The Size of a Set DEFINITION 4 Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by S.

20 The Size of a Set EXAMPLE Let A be the set of odd positive integers less than 10. Then A = 5. Let S be the set of letters in the English alphabet. Then S = 26.

21 The Size of a Set DEFINITION 5 A set is said to be infinite if it is not finite.

22 Power Set DEFINITION 6 Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S).

23 Power Set EXAMPLE What is the power set of the set {0, 1, 2}?

24 Cartesian Products DEFINITION Let A and B be sets. The Cartesian product of A and B, denoted by A B, is the set of all ordered pairs (a, b), where a A and b B. Hence, A B = {(a, b) a A b B}.

25 Cartesian Products EXAMPLE What is the Cartesian product of A = {1, 2} and B = {a, b, c}? What is the Cartesian product A B C, where A = {0, 1}, B = {1, 2}, and C = {0, 1, 2}?

26 Find: Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, C = {2, 3, 8, 9}. A B = A - C = B C = A B = A C =

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