Building Mathematical Statements at a Glance

Does any value of s make the statement "s > 5 and s < 2" true?

For the statement to be correct, we need to satisfy both parts. Unfortunately, we can't have a number that's both greater than five and less than two. That means no value of s will make that statement true.

Does any food satisfy, "This food is a pie or it tastes like a blueberry"?

Plenty of foods do. Since the two atoms in the statement are joined by a disjunction, only one part of the overall statement has to be true to make the whole thing true.

For "this food is a pie" we could have the food be any kind of pie (not π). For the second statement, any food that tastes like a blueberry would also work, from blueberry smoothies or ice cream or blueberries themselves. Blueberry pie would satisfy both parts of the statement, so the overall statement would still be true.

Is there any value of x that makes "x < 4 and not(x² < 20)" true?

A conjunction means each part must be true for the whole statement to be true. Let's break it down into atoms again.

Lots of numbers are less than 4. Next.

Not(x² < 20) is a negation of whatever is inside the parentheses. So the negation of "x² is less than 20" would be "x² is not less than 20" or "x² is greater than or equal to 20."

For the entire statement to be true, we need a number that is less than four but greater than 20 when squared. Does such a number exist? You betcha. If we make x = -5, it satisfies both atoms in the statement, and therefore makes the whole statement true.