ELEMENTARY SET THEORY

1. ELEMENTARY SET THEORY

2. Sets • A set is a well-defined collection of objects. • Each object in a set is called an element or memberof the set. • The elements or objects of the set are enclosed by a pair of braces { }.

3. Notations • Capital or uppercase letters are usually used to denote sets while small or lowercase letters denote elements of a set. •  denotes “is an element of” or “belongs to” •  denotes “is not an element of” or “does not belong to”

4. Example • Let A – the set of letters in the English alphabet B – the set of primary colors C – the set of positive integers • g  Aorange  B100  C

5. Ways of Describing a Set • List (or roster) method describes a set by enumerating the elements of the set. A = {a, b, c, d,…,z} B = {red, yellow, blue} C = {1, 2, 3, 4,…} • Rule (or set builder) method describes a set by a statement or a rule. A = {x|x is an English alphabet} B = {a|a is a primary color} C = {y|y is a positive integer}

6. Definition of Terms • The cardinality of a set A, or the cardinal number of A, denoted as n(A), is the number of elements in A. n(A) = 26, n(B) = 3, n(C) =  • A set is finite if there is one counting number that indicates the total number of elements in the set. A and B are finite sets. • A set is infinite if in counting the elements, we never come to an end. C is an infinite set.

7. Definition of Terms • The null set or empty set, denoted by the symbol , is the set that contains no elements, that is, A is empty iff n(A) = 0. D = {x|x is a month in the Gregorian calendar with less than 28 days} n(D) = 0, so D =  • A singleton set is a set that contains only one element, that is, B is a singleton set iff n(B) = 1. E = {y|y is prime number, 5 < y < 10}

8. Definition of Terms • Sets A and B are equal if they have the same elements. • Set A is a subset of B, denoted as A  B, iff every element of A is also an element of B.Laws of subset: • Every set is a subset of itself, that is, A  A, for any set A. • The null set is a subset of any set, that is,   A, for any set A.

9. Definition of Terms • Example: Subset G = {x|x is an integer} F = {y|y is a whole number} C = {z|z is a positive integer} • C  G because every element of C is found in G.F  C because 0F but 0C.

10. Definition of Terms • Set A is a proper subset of B, denoted AB, if A is a subset of B and there is at least one element of B that is not in A. That is, AB iff AB and AB. P = {1, 3, 5, 7} Q = {3, 7} Then, QP • The set containing all of the elements for any particular discussion is called the universal set, denoted as U.

11. Exercise • Describe the following sets using the list method and give the set cardinality: • A = {x|x is a natural number which is 1 less than a multiple of 3} • B = {a|a is a rational number whose value is 2/3} • C = {b|b is a vowel that appears in the phrase “set of vowels”} • D = {z|z is an even prime integer greater than 2}

12. Exercise • Describe the following sets using the rule method: • F = {0, 1, 8, 27, 64, 125, 216, …} • G = {…, -30, -20, -10, 0, 10, 20, 30, 40, …} • H = {1, 2, 3, 5, 7, 11, 13, 17, 19, 23, …}

13. Exercise • Determine which of the following statements are true and which are false: •   N • {1, 2, 3}  N • {0}  N • {1, 2, 3}  {1, 2, 3} • 1  {1, 2, 3} • n(N) = 1010 • A  B  n(A) > n(B) • {3}  N

14. The Power Set • 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) or (S). • Example: S = {0, 1, 2}(S) = {, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}} • Note that the empty set and the set itself are members of this set of subsets.

15. The Power Set • If a set has n elements, then its power set has 2n elements. • Exercise: • What is the power set of A = {x, y, z}? • What is the power set of the null set? • What is the power set of the power set of the null set? • What is the power set of B = {0, {1}, 3, {2, 4}}

16. SET OPERATIONS

17. Union • Let A and B be sets. The union of the sets A and B, denoted by A  B, is the set that contains those elements that are either in A or in B, or in both. • A  B = {x|x  A  x  B} • Example: A = {1, 3, 5} and B = {1, 2, 3}A  B = {1, 2, 3, 5}

18. Union • Venn Diagram

19. Intersection • Let A and B be sets. The intersection of the sets A and B, denoted by A  B, is the set containing those elements in both A and B. • A  B = {x|x  A  x  B} • Example: A = {1, 3, 5} and B = {1, 2, 3}A  B = {1, 3}

20. Intersection • Venn Diagram

21. Disjoint • Two sets are called disjoint if their intersection is the empty set. • A  B = { } • Example: A = {1, 3, 5, 7, 9} B = {2, 4, 6, 8, 10}A  B = , then A and B are disjoint

22. Difference • Let A and B be sets. The difference of A and B, denoted by A – B, is the set containing those elements that are in A but not in B. • The difference of A and B is also called the complement of B with respect to A. • A – B = {x|x  A  x  B} • Example:A = {1, 3, 5} and B = {1, 2, 3} A – B = {5}

23. Difference • Venn Diagram

24. Complement • Let U be the universal set. The complement of the set A, denoted by A’, is the complement of A with respect to U. In other words, the complement of set A is U – A. • A’ = {x|x  A} • Example: A = {a, e, i, o, u} where the universal set is the set of letters in the English alphabet A’ = {y|y is a consonant}

25. Complement • Venn Diagram

26. Cartesian Products • The order of elements in a collection is often important. Since sets are unordered, a different structure is needed to represent ordered collections. This is provided by ordered n-tuples. • The ordered n-tuple (a1, a2,…an) is the ordered collection that has a1 as its first element, a2 as its second element,…, and an as its nth element. • Two ordered n-tuples are equal iff each corresponding pair of their elements is equal.

27. Cartesian Products • Let A and B be sets. The Cartesian product of A and B, denoted by A x B, is the set of all ordered pairs (a, b) where a  A and b  B. • A x B = {(a, b)|a  A  b  B} • Example: A = {1, 2} and B = {a, b, c} A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)} • Note: A x B  B x A

28. Cartesian Products • The Cartesian product of sets A1, A2,…, An, denoted by A1 x A2 x … x An is the set of ordered n-tuples (a1, a2,…, an), where ai belongs to Ai for i = 1, 2,…, n. • A1 x A2 x … x An = {(a1, a2, …, an) | ai Ai for i = 1, 2, …, n} • Example: A = {0, 1}, B = {1, 2}, C = {0, 1, 2}, find A x B x C.

29. Exercise • Let A be the set of students who live one km from school and let B be the set of students who walk to classes. Describe the students in each of these sets. • A  B • A  B • A – B • B – A

30. Exercise • Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find • A  B • A  B • A – B • B – A • Let A = {0, 2, 4, 6, 8, 19}, B = {0, 1, 2, 3, 4, 5, 6} and C = {4, 5, 6, 7, 8, 9, 10} a. A  B  C b. A  B  C c. (A  B)  C d. (A  B)  C

31. Properties of Set Union • For any sets A, B, and C, • The union of any set with the null set is the set itself. A   = A • The union of any set with itself is the set itself. A  A = A • Set Union is commutative. A  B = B  A • Set Union is associative. (A  B)  C = A  (B  C) • Any set is a subset of its union with another set. A  A  B

32. Properties of Set Intersection • For any sets A, B, and C, • The intersection of any set with the null set is the null set. A   =  • The intersection of any set with itself is the set itself. A  A = A • Set Intersection is commutative. A  B = B  A • Set Intersection is associative. (A  B)  C = A  (B  C) • The intersection of any given set with another set is a subset of the given set. A  B  A • A  (B  C) = (A  B) (A  C)A  (B  C) = (A  B) (A  C)

33. Properties of Set Difference • For any sets A, B, and C, • The removal of the null set from any set has no effect on the set. A –  = A • The removal of the elements of any set from itself will leave the empty set. A – A =  • No elements can be removed from the null set.  – A =  • The result of removing the elements of a set from any given set is a subset of the given set. A – B  A • A – (B  C) = (A – B)  (A – C)A – (B  C) = (A – B)  (A – C)

34. Properties of Set Cardinality • For any sets A, B, and C, • |A  B| = |A| – |A – B| • |A  B| = |A| + |B| – |A  B| • |A – B| = |A| – |A  B|