Question number 14. Consider the following proposition about a function f on the set of integers. For any even x, f of x is even. What would be the negation of this proposition? The proposition we are considering consists of a statement and a universal quantifier. The statement p is that f of x is even. And as a mathematical statement, its truth value is either true or false. Now, the universal quantifier together with the statement makes the proposition in the following form. For all x, the statement p is true. And negated, the statement becomes the negation of the quantifier. So instead of for all, we would say there exists an x such that p is false. Now, thinking about that, there exists an x would mean that there exists an even x, an even number, an even integer. And that p is false means that it's not the case that f of x is even. In other words, it is the case that f of x is odd. And so the only possibility here is statement d. There exists an even number x such that f of x is odd. And that's the correct negation of our statement.

negation

universal quantifier

even integers

mathematical proposition

function