Show that the two sentences below are logically equivalent. Express each pair of sentences using a logical expression. Then prove whether the two expressions are logically equivalent. Note: you can assume that x and y are real numbers, so if x is not irrational, then x is rational, and if x is not rational, then x is an irrational number. If x is a rational number and y is an irrational number then x-y is an irrational number. If x is a rational number and x-y is a rational number then y is a rational number.

Expression 1) : if x is a rational number and y is an irrational number, then x-y is irrational

Expression 2) : If x is rational and x-y is also rational, then y is rational

Lets prove that 1 implies 2:

Lets assume the hypotheses of 2). We have a rational number x and another number y. We want to know, given that x-y is rational, if y is rational or not. If y were not to be rational, that we have a rational number x and an irrational number y; expression 1 says therefore that x-y is irrational, which cant be true. As a consecuence, y cant be irrational, thus, it has to be rational.

Lets prove that 2 implied 1:

Now lets assume that we have a rational number x and an irrational number y. We want to preove that x-y is irrational. If that is not the case, then x-y has to be rational, howver, since x is also rational, we have, for proposition 2, that y has to be irrational, which cant be true because we assume the contrary. As a consecuence, x-y cant be rational, thus, it is irrational.