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Quantifiers

Sentences involving the phrases For every ... and There exists ... also play a very important role in the structure of mathematical sentences. The symbol tex2html_wrap_inline11940 , called the universal quantifier , denotes phrases such as For each, For every, For all. A sentence such as

For every x, P(x)

can be translated symbolically into
displaymath11934

The following sentences have the same meaning:

, x is an integer ,
For every x, if x is an integer, then ,
For all x, if x is an integer, then ,
For each x, if x is an integer, then ,
Every integer belongs to ,
Every integer is a rational number,
If x is an integer, then .

Note that in the last sentence the universal quantifier is understood and not written.

The symbol tex2html_wrap_inline11976 , called the existential quantifier symbolizes phrases such as There exists, There is at least one, For at least one, and Some. A sentence such as

There exists an x such that P(x)

translates symbolically to
displaymath11935

The following have the same meaning:

, x is a natural number.
There exists an x such that x is a natural number.
Some number is a natural number.
There is at least one natural number.

Quantifiers often appear together. Consider the following examples.

tex2html_wrap_inline11990 For every x and for every y, x+y=0.

tex2html_wrap_inline11998 For every x there exists a y so that x+y=0.

tex2html_wrap_inline12006 There exists an x such that for all y, x+y=0.

tex2html_wrap_inline12014 There exists an x and there exists a y such that x+y=0

The following sentence

For every x, if x is even, then there exists a y such that x=2y.

translates as
displaymath11936

These quantifiers refer to some universal set, which if not explicitly given, must be easily inferred from the context. We will be interested only in nonempty universal sets.

Definition: The sentence tex2html_wrap_inline12030 is true if and only if the solution set of P(x) equals the universal set. This sentence is false if the solution set is a proper subset of the universal set; i.e., if there is at least one element of the universal set for which P(x) is false.

Definition: The sentence tex2html_wrap_inline12036 is true if the solution set of P(x) is nonempty. This sentence is false if the solution set of P(x) is empty; i.e., if for every replacement of x by a member a of the universal set, P(a) is false.

For more complicated mathematical sentences containing more quantifiers let us look at a few examples. Example: Suppose P(x,y) is a sentence with variables x and y. The sentence tex2html_wrap_inline12054 is true if and only if for every replacement of x and y by members a and b from the universal set, the statement P(a,b) is true. The sentence is false if there is a replacement for x or a replacement for y for which the statement is false.

Example: The sentence tex2html_wrap_inline12070 is true if there exists a replacement a for x such that tex2html_wrap_inline12076 is true. This same a makes the sentence P(a,b) true for every b in the universal set. Note that the sentence tex2html_wrap_inline12084 is false. There is no replacement a for x which makes the sentence tex2html_wrap_inline12090 true.

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droyster@math.uncc.edu

	For every x and for every y, x+y=0.
	For every x there exists a y so that x+y=0.
	There exists an x such that for all y, x+y=0.
	There exists an x and there exists a y such that x+y=0