Section 17.3 Quantified Statements
Objectives
Identify and evaluate logical statements involving universal and existential quantifiers.
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.
Definition 17.3.1. 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 17.3.2.
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 17.3.3.
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?
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.
Theorem 17.3.4. Negating a quantified statement.
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 17.3.5.
“Somebody brought a flashlight.” Write the negation of this statement.
The negation is “Nobody brought a flashlight.”
Example 17.3.6.
“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 17.3.7.
Write the negation of “All Icelandic children learn English in school.”
The negation is “At least one Icelandic child does not learn English in school.”