3.2.3: Quantified Statements
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3.2.3 Learning Objectives
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Interpret the use of universal quantifiers and existential quantifiers correctly
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Negate a quantified statement
Words that describe an entire set, such as “all”, “every”, or “none”, are called universal quantifiers because the description applies to all elements in the 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: 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 with that characteristic.
Something interesting happens when we negate – or state the opposite of – a quantified statement.
Example 1
Suppose your friend says “Everybody cheats on their taxes.” This statement uses a universal quantifier. 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 2
Suppose your friend says “One of these six cartons of milk is leaking.” This statement uses an existential quantifier. 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 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 3
“Somebody brought a flashlight.” Write the negation of this statement.
The negation is “Nobody brought a flashlight.”
Example 4
“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
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At least one Icelandic child did not learn English in school.