1.4 Sets

1. 1.4 Sets Example 2 The set ofpositive integers less than 100 can be denoted as Example 3 A set can also consists of seemingly unrelated elements: Definition 1. A set is a group of objects．The objects in a set are called the elements, or members, of the set. Definition 2. Two sets are equal if and only if they have the same elements.

2. Example 6 Draw a Venn diagram that presents V, the set of vowels in English alphabet. U a,e,i,o,u V • A set can be described by using a set builder notation. • A set can be described by using a Venn diagram.

3. The set that has no elements is called empty set, denoted by . Definition 3. The set A is said to be a subset of B if and only if every element of A is also an element of B. We use the notation to indicate that A is a subset of the set B. U A B

4. Example 10 The set of positive integers is infinite. 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 n is the cardinality of S. The cardinality of S is denoted by |S|. Definition 5. A set is said to be infinite if it is not finite.

5. Cartesian Products The Power Set Definition 5. The power set of a set S is the set of all subsets of S, denoted by P(S).

7. Definition 1. Let A and B be sets. The union of the sets A and B, denoted by , is the set that contains those elements that are either A and B, or in both. That is Definition 2. Let A and B be sets. The intersection of the sets A and B, denoted by , is the set that contains those elements that are in both A and B. That is U U A B B 1.5 Set Operations

8. Definition 4. Let A and B be sets. The difference of A and B, denoted by A-B is the set containing of 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. That is, U A B Definition 3．Two sets are called disjoint if their intersection is the empty set.

9. Definition 5. Let U be the universal set. The complement of the set A, denoted by , is the complement of A with respect to U. In other words, the complement of the set is U-A. That is, U A

10. Set Identities Table 1 Set Identities Identity Name Identity laws Domination laws Idempotent laws Complementation laws Commutative laws Associative laws Distributive laws De Morgan’s law

11. One way to prove that two sets are equal is to show that one of sets is a subset of the other and vise versa. • One way to prove that two sets are equal is to use set builder and the rules of logic.

12. Table 2. A membership table for the distributive Property A B C 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 • Set identities can be proved by using membership tables. • Set identities can be established by those that we have already proved.

13. Definition 1. Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a)=b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f : A B. x x y Example 1 Let set A={Adams, Chou, Goodfriend, Rodriguez, Stevens} and B={A,B,C,D,F}. Let G be the function that assigns a grade to a student in our discrete mathematics. z A function Not a function Adames Chou Goodfriend Rodriguez Stevens A B C D F G 1.6 Functions The domain of G is the set A={Adams, Chou, Goodfriend, Rodriguez, Stevens}, and the range of G is the set {A,B,C,F}.

14. f b=f(a) a A B f Definition 2. If f is a function from A to B, we say that A is the domain of f and B is the codomain of f. If f(a)=b, we say that b is the image of a and a is a pre-image of b. The range of f is the set of all images of elements of A. Also, if f is a function from A to B, we say that f maps A to B. The domain and codomain of f is Z, and the range of f is the set {0,1,4,9,…}.

15. Definition 4. Let f be a function from the set A to the set B and let S be a subset of A. The image of S is the subset of B that consists of the images of elements of S. We denote the image of S by f(S), so that Example 4 Let A={a,b,c,d,e} and b={1,2,3,4} with f(a)=2, f(b)=1,f(c )=4, f(d)=1, and f(e)=1. The image of S={b,c,d} is the set f(S)={1,4}. A S f(S) B

16. One-to-One and Onto Functions ｘ ｙ ｘ ｙ ｆ（ｘ）　　ｆ（ｙ） ｆ（ｘ） ｆ（ｙ） Example 6 Determine whether the function f from {a,b,c,d} to {1,2,3,4,5} with f(a)=4, f(b)=5, f(c )=1, f(d)=3 is one to one. one-to-one function function but not one-to-one 1 a b c d 2 3 4 5 Definition 5. A function is said to be one-to-one, or injective, if and only if f(x)=f(y) implies that x=y for all x and y in the domain of f. A function is said to be an injection if it is one-to-one.

17. Example 8 Determine whether the function f from {a,b,c,d} to {1,2,3} with f(a)=3, f(b)=2, f(c )=1, f(d)=3 is onto. B A into 1 a b c d B 2 A 3 onto Definition 6.A function f whose domain and codomain are subsets of the set of real numbers is called strictly increasing if f(x)<f(y) whenever x<y and x and y are in the domain of f. Similarly, f is called strictly decreasing if f(x)>f(y) whenever x<y and x and y are in the domain of f. • A strictly increasing or strictly decreasing function must be one-to-one.

18. A B A B ｘ ｙ ｆ（ｘ）　　ｆ（ｙ） one-to-one onto a 1 a 1 a 1 a 1 a 2 1 2 b b 2 b 2 2 3 3 b b c 3 c 3 c 3 4 4 c c e 4 4 e one-to-one and onto not a function one-to-one, not onto onto, not one-to-one neither one-to-one nor onto Definition 8．The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto. + Example 10

19. Inverse Function and Compositions of fuctions f B A f Example 11 Let f be the function from the set of integers to the set of integers such that f(x)=x+1. Is f invertible, and if it is, what is its inverse? • A function is invertible if it is one-to-one correspondence, and it is not invertible if it is not one-to-one correspondence.

20. Definition 10．Let g be a function from set A to the set B and let f be a function from the set B to set C. The composition of the functions f and g, denoted by f g, is defined by f g a A f(g(a)) C g(a) B g f Example 13 Let f and g be the functions from the set of integers to the set of integers defined by f(x)=x+3 and g(x)=3x+2. What are

21. Let f be a one-to-one correspondence function from set A to set B and • be the inverse of f. Some Important Functions • Other Functions • Polynomial functions • logarithmic functions • exponential functions

22. Definition 1. A sequence a is function from a subset of the set of integers to a set S. We use the notation to denote the image of the integer n. We call a term of the sequence. 1 2 n 1.7Sequences and Summations

23. Special Integer Sequences Example 3. What is a rule that can produce the terms of a sequence if the first 10 terms are 1,2,2,3,3,3,4,4,4,4? Example 4. What is a rule that can produce the terms of a sequence if the first 10 terms are 5,11,17,23,29,35,41,47,53,59? Finding a formula or a general rule for constructing the terms of a sequence. • Are there are runs of the same value? • Are terms obtained from previous terms by adding or multiplying a particular amount? • Are the terms obtained by combining previous terms in a certain way? Solution: A reasonable guess is that the nth term is 5+6(n-1)=6n-1.

24. Summations