Question

Discrete Mathematics

Question 1:
(a) Use the method of generalizing from the generic particular in a direct proof to show that the sum of any two odd integers is even. See the example on page 152 (4th edition, Discrete Mathematics with Applications) for how to lay this proof out.
(b) Determine whether 0.151515... (repeating forever) is a rational number. Give reasoning.
(c) Use proof by contradiction to show that for all integers n, 3n + 2 is not divisible by 3. (d) Is {{5,4},{7,2},{1,3,4},{6,8}} a partition of {1,2,3,4,5,6,7,8}? Why?

Answer 1

Answer 1: such Z is an even number. (a) Let any y be two odd integers that n + y = 2 Now, we need to proof that Since x is odd, x= akt1 for some integer And is also odd, anti, for some integer k n Then , z = ht + y = 2kti + anti - ak tanta 2 (k+n+1) ; where j = (k+n+D is some integer 2 .. Since, z= iz is an even number Hence, proved that the sum of two odd integers is even. 2 (b) We need to determine whether 0.151515..... is a rational number. For that we need to convert 0.1515 in the form of Pla Now, 15 = 0.15 ISIS...ve 0.15 99 15 o.is ga 99 ; which satisfies the definition of a rational number. Hence, o.1515.... is a rational number. =

3 (c) To proof. It n is an integer, then ant? is not divisible by Proof Let us assume that n is an integer such that 3n+ 2 is divisible by 3. Since 3n+2 is divisible by 3, there exists integer m, such that unta = 3m an integer

3n+2=3m 3m 3n 3 (mon) -Y their are integers is also an integer. 3. Since and n difference mon Since 2 is equal to 3 multiplied by some integer, the definition of divisible tells us that a es divisible by There exists a contradiction , because 3 doesn't divide a. Thus, the assumption, we had taken is false, which implies that 3nta is nof divisible by 3 for any integer Hence, proved.

(d) In the set {{5,4} , {1, 2} , { 1, 3, 44, 46,8}}, 4 is repeated in two sets i. e, in &5, uy & {1,3,4} But, there is only {1,2,3,4,5,6,7,8} one 4 in Thus, the answer is no, {{5,4} {1,2}, {1,3,4},{6,8}} is not partition of {1,2,3,4,5,6,7,8}