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Remarks. We do not describe the sets
Quantifiers. Quantifiers are important mathematical tools. Using quantifiers, we can make our mathematical language precise. There are two principal quantifiers in mathematics: the universal and the existential.
Universal quantifier. The universal quantifier has the symbolic form ∀. To express the universal quantifier in English, we write “for all,” “for every,” or “for each.”
Examples (1.5) The following statements are equivalent.
1. (∀x ∈
2. For every x in
3. For every natural number x, the number x + 1 is also a natural number.
4. Given any natural number x, x + 1 is a natural number.
Remark. Of course, there are other ways to state the sentence: “For all x ∈
Existential quantifier. The existential quantifier has the symbolic form ∃. To express the existential quantifier in words, we say “there exists” or “there is” or “for some” or “there is at least one.”
Examples (1.6) The following statements axe equivalent.
1. (∃x ∈
2. There exists a natural number x such that x > 5.
3. There is at least one natural number greater than 5.
4. For some natural number x, the number x is greater than 5.
The phrase “there exists [some object]” is often followed by “such that.” The phrase “such that” is used in mathematics instead of phrases involving the relative pronouns “which,” “that,” or “whose.”
Order of quantifiers. The order of quantifiers in a sentence is important. The following examples illustrate this point.
Examples (1.7) Consider these two statements:
1. (∀x ∈
For each x ∈
2. (∃y ∈
There exists y ∈
Statement 1 says that given any positive integer, there is a larger positive integer. Statement 1 is true.
Since “∀x ∈
Let x = 5. There exists y ∈
Let x = 6. There exists y ∈
Statement 2 says that there is a positive integer which is larger than every positive integer, including itself.
Statement 2 is false.
Since “∃y ∈
Exercises (1.5) Write out each statement using words rather than symbols. Then classify the statements either true or false. Explain your answers.
1. (∃x ∈
2. (∀y ∈