Question

(d) For each of the following, prove or disprove: iii) There is an element of X × Y with the form (a, 3a)

Prove or Disprove #3

Answer 1

(iii). False.

Suppose there exist an element of the form ( a , 3a ) in X × Y .

$\Rightarrow$ a $\in$ X and 3a $\in$ Y .

Since 3a $\in$ Y and Y = { y : y = 3m + 1 for some m $\in$$\mathbb{Z}$

$\Rightarrow$ 3a = 3m + 1

$\Rightarrow$ 3a - 3m = 1

$\Rightarrow$ 3 ( a - m ) = 1

Here left hand side is a multiple of 3 where right hand side is not , a contradiction

Hence there does not exist any element of the form (a , 3a ) in X × Y

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.If you have any doubt or need more clarification at any step please comment.