Answer
Answer is given below
Work Step by Step
It is easiest to give proofs by contraposition of ( i) --+ (ii), (ii) --+ ( i), ( i) --+ (iii) , and (iii) --+ ( i). For the
first of these, suppose that 3x + 2 is rational, namely equal to p / q for some integers p and q with q -=I- 0.
Then we can write x = ( (p/ q) - 2) /3 = (p - 2q) / (3q), where 3q -=I- 0. This shows that x is rational. For
the second conditional statement, suppose that x is rational, namely equal to p/q for some integers p and q
with q -=I- 0. Then we can write 3x + 2 = (3p + 2q) / q, where q -=I- 0. This shows that 3x + 2 is rational. For
the third conditional statement, suppose that x/2 is rational, namely equal to p/q for some integers p and q
with q -=I- 0 . Then we can write x = 2p / q, where q -=I- 0 . This shows that x is rational. And for the fourth
conditional statement, suppose that xis rational, namely equal to p/q for some integers p and q with q -=I- 0.
Then we can write x/2 = p/(2q), where 2q -=I- 0. This shows that x/2 is rational.
