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Susanna S. Epp DISCRETE MATHEMATICS WITH APPLICATIONS FOURTH EDITION
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, b Є Q, If r is irrational and b≠ 0, then a+br is irrational
Let us begin the proof Let the negative be true a, b Є Q, 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, b Є Q m/n + (f/g)r = x/y – m/n (f/g)r = (xn – mg)/(yn) r = (gxn – gmy)/(fyn) r Є Q since it can be written the form a/b
Conclusion This contradicts that r is irrational. Hence the original statement is true. Proof by contradiction.
Section 9. 2 #18 18. The diagram below shows the keypad fora. How many different PINs are represented by the an automatic same sequence of keys as teller machine. As you can see, the same 2133? sequence of keys represents a variety of different PINs. b. How many different PINs For instance, are represented by the 2133, AZDE, and BQ 3 F are all keyed in same sequence of keys as exactly the 5031? same way. 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
a. 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.
c. Constructing a numeric PIN with no repeated digit can be thought of as the following fourstep 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 of numeric PINs with no repeated digit is 10 * 9 * 8* 7 = 5040.
Blaise Pascal Born June 19, 1623 Clermont-Ferrand, France Died August 19, 1662 Paris, France
Biography “Pascal, Blaise (1623 -62), French philosopher, mathematician, and physicist, considered one of the great minds in Western intellectual history. Pascal was born in Clermont-Ferrand on June 19, 1623, and his family settled in Paris in 1629. Under the tutelage of his father, Pascal soon proved himself a mathematical prodigy, and at the age of 16 he formulated one of the basic theorems of projective geometry, known as Pascal's theorem and described in his Essai pour les coniques (Essay on Conics, 1639). In 1642 he invented the first mechanical adding machine. Pascal proved by experimentation in 1648 that the level of the mercury column in a barometer is determined by an increase or decrease in the surrounding atmospheric pressure rather than by a vacuum, as previously believed. This verified the hypothesis of the Italian physicist Evangelista Torricelli concerning the effect of atmospheric pressure on the equilibrium of liquids. Six years later, with the French mathematician Pierre de Fermat, Pascal formulated the mathematical theory of probability, which has become important in such fields as actuarial, mathematical, and social statistics and as a fundamental element in the calculations of modern theoretical physics. Pascal's other important scientific contributions include the derivation of Pascal's law or principle, which states that fluids transmit pressures equally in all directions, and his investigations in the geometry of infinitesimals. His methodology reflected his emphasis on empirical experimentation as opposed to analytical, a priori methods, and he believed that human progress is perpetuated by the accumulation of scientific discoveries resulting from such experimentation. “ Source: http: //library. thinkquest. org/10170/voca/pascal. htm#
CHAPTER 9. 5 #7 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 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) .