Discrete Mathematics with Applications Fourth Edition

1. Susanna S.Epp Discrete Mathematics with ApplicationsFourth Edition

2. Section 4.6 Problem #12 • If a and b are rational numbers, b≠0, and r is an irrational number, then a+br is irrational. Let us first rewrite this problem in a more mathematical format. ⋁a,bЄQ, If r is irrational and b≠0, then a+br is irrational

3. Let us begin the proof Let the negative be true a,bЄQ, r is rational, b≠0 and a+br is rational a+br = x/y by def of rational numbers Let a=m/n and b=f/g since a,bЄQ m/n + (f/g)r = x/y (f/g)r = x/y – m/n (f/g)r = (xn – mg)/(yn) r = (gxn – gmy)/(fyn) r ЄQ since it can be written the form a/b

4. Conclusion This contradicts that r is irrational. Hence the original statement is true. Proof by contradiction.

5. Section 9.2 #18 How many different PINs are represented by the same sequence of keys as 2133? b. How many different PINs are represented by the same sequence of keys as 5031? c. At an automatic teller machine, each PIN corresponds to a four-digit numeric sequence. For instance, TWJM corresponds to 8956. How many such numeric sequences contain no repeated digit 18. The diagram below shows the keypad for an automatic teller machine. As you can see, the same sequence of keys represents a variety of different PINs. For instance, 2133, AZDE, and BQ3F are all keyed in exactly the same way.

6. Let step 1 be to choose either the number 2 or one of the letters corresponding to the number 2 on the keypad, let step 2 be to choose either the number 1 or one of the letters corresponding to the number 1 on the keypad, and let steps 3 and 4 be to choose either the number 3 or one of the letters corresponding to the number 3 on the keypad. There are 4 ways to perform step 1, 3 ways to perform step 2, and 4 ways to perform each of steps 3 and 4. So by the multiplication rule, there are 4*3*4*4 = 192 ways to perform the entire operation. b. Constructing a PIN that is obtainable by the same keystroke sequence as 5031 can be thought of as the following four-step process. Step 1 is to choose either the digit 5 or one of the three letters on the same key as the digit 5, step 2 is to choose the digit 0, step 3 is to choose the digit 3 or one of the three letters on the same key as the digit 3, and step 4 is to choose either the digit 1 or one of the two letters on the same key as the digit 1. There are four ways to perform steps 1 and 3, one way to perform step 2, and three ways to perform step 4. So by the multiplication rule there are 4 *1 * 4 * 3 = 48 different PINs that are keyed the same as 5031.

7. c. Constructing a numeric PIN with no repeated digit can be thought of as the following four-step process. Steps 1-4 are to choose the digits in position 1-4 (counting from the left). Because no digit may be repeated, there are 10 ways to perform step one, 9 ways to perform step two, 8 ways to perform step three, and 7 ways to perform step four. Thus the number ofnumeric PINs with no repeated digit is 10 * 9 * 8* 7 = 5040.

9. Biography

10. . A computer programming team has 13 members. a.)How many ways can a group of seven be chosen to work on a project? (13 C 7) b.) Suppose seven team members are women and six are men. (i) How many groups of seven can be chosen that contain four women and three men? (7 C 4)*(6 C 3) (ii) How many groups of seven can be chosen that contain at least one man? (13 C 7) – (7 C 7) (iii) How many groups of seven can be chosen that contain at most three women? (7 C 3)*(6 C 4) + (7 C 2)*(6 C 5) + (7 C 1)*(6 C 6) c.) Suppose two team members refuse to work together on projects. How many groups of seven can be chosen to work on a project? (11 C 6) + (11 C 6) + (11 C 7) d.) Suppose two team members insist on either working together or not at all on projects. How many groups of seven can be chosen to work on a project? (11 C 7) + (11 C 5) Chapter 9.5 #7