I am a freshman college student and I am new to discrete mathematics. I tried to look for something in my textbook to help me figure out a problem I have but I don't really understand how to solve it. Here is the problem:
Prove by contradiction that for all x, y ∈ R, if x is rational and y is irrational, then x+y is irrational.
I'm not really sure what to do. Could I please get some help? Thank you!
You are asked to prove "if x is rational and y is irrational, then x+y is irrational". It seems strange that you would be asked to use a "proof by contradiction" if you don't know what that means- but that seems to be what you are saying. A "proof by contradiction" starts by denying the conclusion, then show that leads to a contradiction. You want to prove "x+ y is irrational" so start by asserting that "x+ y is rational". Then y= (x+y)- x is the difference of two rational numbers. What does that tell you about y?
Correct. First you assume the premises: that x is rational and y is irrational. Then you assume the negation of the conclusion, i,e., you assume that x + y is rational. This implies that y = (x + y) - x is rational as the difference of two rational numbers. This contradicts one of the assumptions. Therefore, the assumption "x + y is rational" is impossible and x + y is, in fact, irrational.