17.4: Quantified Statements

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Words that describe an entire set, such as “all”, “every”, or “none”, are called universal quantifiers because that set could be considered a universal set. In contrast, words or phrases such as “some”, “one”, or “at least one” are called existential quantifiers because they describe the existence of at least one element in a set.

Quantifiers

A universal quantifier states that an entire set of things share a characteristic.

An existential quantifier states that a set contains at least one element.

Something interesting happens when we negate – or state the opposite of – a quantified statement.

Example 11

Suppose your friend says “Everybody cheats on their taxes.” What is the minimum amount of evidence you would need to prove your friend wrong?

To show that it is not true that everybody cheats on their taxes, all you need is one person who does not cheat on their taxes. It would be perfectly fine to produce more people who do not cheat, but one counterexample is all you need.

It is important to note that you do not need to show that absolutely nobody cheats on their taxes.

Example 12

Suppose your friend says “One of these six cartons of milk is leaking.” What is the minimum amount of evidence you would need to prove your friend wrong?

Solution

In this case, you would need to check all six cartons and show that none of them is leaking. You cannot disprove your friend’s statement by checking only one of the cartons.

When we negate a statement with a universal quantifier, we get a statement with an existential quantifier, and vice-versa.

Negating a quantified statment

The negation of “all A are B” is “at least one A is not B”.

The negation of “no A are B” is “at least one A is B”.

The negation of “at least one A is B” is “no A are B”.

The negation of “at least one A is not B” is “all A are B”.

Example 13

“Somebody brought a flashlight.” Write the negation of this statement.

The negation is “Nobody brought a flashlight.”

Example 14

“There are no prime numbers that are even.” Write the negation of this statement.

The negation is “At least one prime number is even.”

Try it Now 1

Write the negation of “All Icelandic children learn English in school.”

Answer: At least one Icelandic child did not learn English in school.