Question

State True or False: “Of the two integers, if one is negative then their product must be positive.”\[\]

Answer 1

Hint: We assign the first part of the given statement “One of the two integers is negative. ” as $p$ and the second part “The product of the two integers is positive. ” as $q$. The given statement becomes $p\Rightarrow q$ whose truth value we check by taking $p$ as true and then checking whether $q$ is false. \[\]

Complete step by step answer:
We know from mathematical logic that a statement is a declarative sentence to which we can assign a truth value either true T or false F.\[\]
The given statement is “Of the two integers, if one is negative then their product must be positive.” This is an implicative statement of the type $p\Rightarrow q$. The first sentence $p$ is the antecedent or premise which is “One of the two integers is negative. ” and the second statement $q$ is the consequence or result, “The product of the two integers is positive. ” The composite statement $p\Rightarrow q$ will be false when the premise $p$ is true and the result $q$ is false.\[\]
So let us assume the statement $p$ is true. So we take two numbers say $a,b$ and check the truth value of $q$. Now we have two cases\[\]
Case-1: There is at least one of the two numbers which is negative. So let us take $a$ as the negative number, in symbols $a < 0$. We know that the product if a negative and a positive number is always negative. So we have $a\times b<0$. We illustrate it by taking two numbers say $-2,3$ and then multiplying to get the product as $\left( -2 \right)\left( 3 \right)=-6$. So the statement is false for case-1.\[\]
Case-2:Both the numbers are negative. So we have $a < 0, b < 0$. We know that the product of two negative numbers is always positive. So we have $a\times b > 0$. So the statement is true for case-2.\[\]

So the whole statement is false.

Note: We need to be careful of the distinction between the connective “if ...then....” $p\to q$ and the implication $p\Rightarrow q$. The latter implication is a type of connective “if... then...” $p\to q$ when $p\to q$ is always true. We note that there may not be any relation between $p,q$ in $p\to q$ but in $p\Rightarrow q$ there is a causal relation of $q$ from $p$.