What is the contrapositive of this simple proposition?

Question

If $a$ and $b$ are rational then $ab$ and $a+b$ are also rational.

What is the contrapositive of this proposition?

1. If $ab$ and $a+b$ are not rational then $a$ and $b$ are not rational.

2. If $ab$ or $a+b$ are not rational then $a$ or $b$ are not rational.

Answer 1

It's probably easiest to formulate your statement in symbolic form then interpret: $$(a \in \mathbb{Q} \land b \in \mathbb{Q} ) \Rightarrow (ab \in \mathbb{Q} \land a+b \in \mathbb{Q})$$ therefore the contrapositive is $$\lnot (ab \in \mathbb{Q} \land a+b \in \mathbb{Q}) \Rightarrow \lnot (a \in \mathbb{Q} \land b \in \mathbb{Q} )$$ or equivalently $$(\lnot ab \in \mathbb{Q} \lor \lnot a+b \in \mathbb{Q}) \Rightarrow (\lnot a \in \mathbb{Q} \lor \lnot b \in \mathbb{Q} ).$$

It's a little unclear to which of your statements this corresponds (ambiguity in your English, at least to me), but I think your (b) is correct. That is, I think your (b) means "If $ab$ is not rational or $a + b$ is not rational then $a$ is not rational or $b$ is not rational."

As a side note, I'm never sure if the use of "are" in English (as in the question) implies parentheses. Maybe someone else can chime in to clear up that point for me to make this a more complete answer.

Answer 2

Your statement: $a$ and $b$ rational $\implies$ ($ab$ and $a+b$) rational.

Contrapositive: not ($a+b$ and $ab$ rational) $\equiv$ not ($a+b$ rational) or not ($ab$ rational) $\implies$ not($a$ and $b$ rational) $\equiv$ not ($a$ rational) or not ($b$ rational).

(My initial answer had a silly mistake in it)