Q.Are the following pairs of statements negations of each other:

Every real number x is either rational or irrational, and never both — this is an exhaustive dichotomy. So "x is not rational" $\equiv$ "x is irrational", and "x is not irrational" $\equiv$ "x is rational".

1. Statement A: "The number x is not a rational number" ($\sim p$, i.e., x is irrational). Statement B: "The number x is not an irrational number" ($\sim(\sim p) = p$, i.e., x is rational). $\sim A$ (negation of A) = "x is a rational number" = B. So A and B are negations of each other.
2. Statement A: "The number x is a rational number" ($p$). Statement B: "The number x is an irrational number" ($\sim p$). …

Let p = "x is rational". Rewrite each of the two given statements in terms of p or ¬p using the fact that "not rational" means "irrational" for real numbers. Two statements ar …

• Treating "not rational" and "irrational" as unrelated properties rather than recognizing they are logically identical for real numbers. …