Expert: Sherry Wallin - 4/4/2011

Question
Prove this proposition using contrapositive or contradiction.  For all real numbers t, if t^2 is irrational, then t is irrational.

For each formula, use a graph to find the largest subset of R that can be the domain of a function with that formula, assuming the codomain is R.  For that domain, determine the range.  j(x)=cot(x)=cos(x)/sin(x)

Answer
Heather~

I will do a proof for both ways using contrapositive and contradiction.

contrapositive: this is equivalent to saying if t is rational then so is t^2:
so sps t is rational which means t = p/q where q/=0 and both p,q are integers and coprime.
Now that means that t^2 = p^2/q^2 or that t^2*q^2 = p^2 or (tq) = p. But this implies that t and q are factors of p. For sure q can't be a factor of p because we already established that p and q are coprime (relatively prime, i.e., have no common factors). Thus it must be that t^2 and t are not rational and thus irrational.

Contradiction:  sps t^2 is irrational and that t is rational. Then t = p/q, q/=0 and p and q are coprime. From t = p/q we also have t^2 = (p/q)^2 which is a contradiction since t^2 is irrational and t*t = (p/q)(p/q) is rational.


I'm not certain what the 2nd par of your question is about, it seems to be unrelated. For each formula? What formula?

Math Prof