1. ## rational irrational expressions

Okay I feal so stupid because I cannot figure this out but I am just brain dead and this homework is due in a couple hours so some help would be greatly appreciated...

Note: A real number m is rational iff m=p/q, were p and q are integers and q is not equal to zero.

problem:

Suppose x and y are real numbers, prove or give a counterexample.

a) if x is irrational and y is irrational, then x + y is irrational
b) if x + y is irrational, then x is irrational or y is irrational
c) if x is irrational and y is irrational, then xy is irrational
d) if xy is irrational, then x is irrational or y is irrational.

2. Originally Posted by luckyc1423

a) if x is irrational and y is irrational, then x + y is irrational
No consider. $x=\pi$ and $y=-\pi$.
b) if x + y is irrational, then x is irrational or y is irrational
Assume that "x is irrational or y is irrational" is false. Meaning "x is rational and y is rational" (if you learned logic this is called de'Morgan's law of negation).
Then, x+y is rational. A contradiction because x+y is irrational. Thus, this statement is true.
c) if x is irrational and y is irrational, then xy is irrational
No. Consider $x=\sqrt{2}$ and $y=\frac{1}{\sqrt{2}}$.
d) if xy is irrational, then x is irrational or y is irrational.
I will leave this as an excerise.