Question

Discrete Mathematics.

(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 165 of the 5th edition of Discrete Mathematics with Applications, Metric Version 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 (a):

Let x and y be two odd integers, such that x + y = z (say).

Now, according to the question, we need to prove that z is an even number, i.e., z = 2*j , for some integer j.

Since x and y are odd, x= 2k + 1, for some integer k and y=2n + 1, for some integer n.

Then, z = x + y = 2k + 1 + 2n + 1 = 2k + 2n +2 = 2( k + n +1 ) = 2j where j = (k + n 1) is some integer

Since, z = 2j , we can say that z is an even number.

Hence, proved that the sum of two odd integers is even.

Answer (b):

According to the definition of a rational number, a number which can be represented in the form p / q is said to be a rational number.

Now, 0.15151515..... = 15 / 99 ; which satisfies the definition of a rational number.

Thus, 0.151515...... is a rational number.

Answer (c):

To prove : If n is an integer, then 3n+2 is not divisible by 3.

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 an interger m, such that 3n + 2 = 3m.

i.e., 3n + 2 = 3m

=> 3m - 3n = 2

=> 3( m - n ) = 2

Since, m and n are integers, their difference m - n is also an integer.

Since 2 is equal to 3 multiplied by some integer, the definition of divisibility tells us that 2 is divisible by 3. But this is not true.

So, there exists a contradiction.

Thus the assumption we took is false, which implies that 3n + 2 is not divisible by 3 for any integer n.

Hence , proved.

Answer (d):

{{5,4} , {7,2} , {1,3,4} , {6,8} } is not a partition of {1,2,3,4,5,6,7,8} because the number 4 is repeated in two subsets of {{5,4} , {7,2} , {1,3,4} , {6,8}} i.e., in {5,4} and {1,3,4}.

Hence, {{5,4} , {7,2} , {1,3,4} , {6,8} } is not a partition of {1,2,3,4,5,6,7,8}.