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

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

4. 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. The set that has no elements is called empty set, denoted by. A B U

5. 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. Example 10 The set of positive integers is infinite.

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

8. 1.5 Set Operations 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 AB U B U

9. Definition 3 ． Two sets are called disjoint if their intersection is the empty set. 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, AB U

10. 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, A U

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

12. 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.

13. Set identities can be proved by using membership tables. 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 1110111011101110 1110000011100000 1100000011000000 1010000010100000 1110000011100000 Set identities can be established by those that we have already proved.

14. 1.6 Functions 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. 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. Adames Chou Goodfriend Rodriguez Stevens ABCDFABCDF G 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}. x y z A function x Not a function

15. 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. f a b=f(a) A B f The domain and codomain of f is Z, and the range of f is the set {0,1,4,9,…}.

16. 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 S f(S) A B 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}.

17. One-to-One and Onto Functions 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. ｘｙｘｙ ｆ（ｘ） ｆ（ｙ） function but not one-to-one ｘｙｘｙ ｆ（ｘ） ｆ（ｙ） one-to-one function 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. abcdabcd 1 2 3 4 5

18. 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. A strictly increasing or strictly decreasing function must be one-to-one. 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. abcdabcd 1 2 3 B A onto A into B

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

20. Inverse Function and Compositions of fuctions f f A B A function is invertible if it is one-to-one correspondence, and it is not invertible if it is not one-to-one correspondence. 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?

21. aAaA g(a) B f(g(a)) C g f 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 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

22. 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

23. 1.7 Sequences and Summations 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. 1n2

24. Special Integer Sequences 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? 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? Solution: A reasonable guess is that the nth term is 5+6(n-1)=6n-1.

25. Summations