Question

If the statement is true, prove it; if it is false, give a counterexample:

1. If a | b and c | d in R, then .
2. If a | b c | d in R, then ( a + c ) | ( b + d )

Short answer

1. The statement is true and has been proved.
2. The statement is false and a counter example has been provided.

Step-by-step solution

Given that

Given that a | b and c | d in R.

If a | b then, b=$k_1$a for some integer .

If c | d then, d=$k_2$c for some integer

Prove that ac | bd

Now, b=$k_1$a and d=$k_2$c

Multiplying both:

bd=( $k_1{k}_2$) ac

Here, $k_1{k}_2$is an integer as $k_1$and $k_2$both are integers.

Hence, ac | bd.

Conclusion

Thus, it can be concluded that if a | b and c | d then, ac | bd.

Take example

Given that a | b and c | d in R.

Take $\mathbf{\mathbb{Z}}$as integral domain.

Let $\mathit{a}\mathbf{=}\mathit{b}\mathbf{=}\mathit{c}\mathbf{=}\mathbf{1}$and$\mathit{d}\mathbf{=}\mathbf{2}$

Prove that the statement is false

Here, 1 divides 1 and 1 divides 2 as well.

That is a | b and c | d

But (1+1) does not divides (1+2), that is, 2 does not divide 3

Conclusion

Thus, it can be concluded that if a | b and c | d then,( a + c) does not divide ( b + d ) .