Theorem: For any two real numbers, x and y, if x and y are both rational then x + y is also rational. Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem?

The contrapositive of the theorem: "for any two real numbers, x and y, if x and y are both rational then x + y is also rational" is given by this:

If x + y is irrational then x is irrational or y is irrational.

What do we mean by contrapositive of a statement?

If we have two statements p and q, such that p ⇒ q, then the contrapositive of this relation is q' ⇒ p', that is negative of q implies a negative of p.

How do we solve the given question?

Let our statements be,

p: x and y are both irrational

∴ p': x is irrational or y is irrational

q: x + y is irrational.

∴ q': x + y is irrational.

The given theorem is the relation, p ⇒ q.

We are asked to find the contrapositive of this theorem. The contrapositive of the relation is, q' ⇒ p'.

This relational statement is:

If x + y is irrational then x is irrational or y is irrational.

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