Prove the following sentences using any prove method [ the most appropriate one]: a) If r is rational and s is irrational, prove that 2r+s is irrational. b) If t and s are integers and t × s is even, then t is even or s is even.

Solution:

(a):

Suppose r is rational and s is irrational. Assume for contradiction that r + 2s is rational. Then there are integers a and b with b $\ne$ 0 such that r + 2s = $\frac ab$ . Since r is rational, there are integers c and d such that r =$\frac cd$ . Then 2s = $\frac ab-\frac cd$ = $\frac{ad−bc}{bd}$ is rational, contradicting the fact that 2s is irrational.

Hence, proved.

(b):

Given that t and s are integers and t × s is even.

So, t × s = 2m, for some constant m.

$\dfrac{t × s }2=m \\\Rightarrow \dfrac{t }2× s=m \ or\ \dfrac{s }2× t=m$

$\Rightarrow$ Either t is divisible by 2, so it is even

or s is divisible by 2, so it is even.

Hence, proved.