The Language of Mathematics - free variables,
connectives and implicit quantifiers
P. Howard
September 12, 2016
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
1 / 22
The words free and bound
For a variable x we refer to the four quantifiers “∀x”, “∀x ∈ A”, “∃x” and
“∃x ∈ A” as quantifiers for x. If P is a sentence in which the variable x
occurs then x is said to be a bound variable in P if, in P, x is in the scope
of a quantifier for x. Otherwise x is a free variable in P. Sometimes the
word unquantified is used instead of the word “free” and the word
quantified is used instead of the word “bound”.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
2 / 22
example
Example
Identify the free variables and the bound variables in the following
sentences. Assume, unless indicated otherwise, that the range of every
variable is the set of real numbers.
1
∀x ∈ R, x 2 + 7x ≥ xy
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
3 / 22
example
Example
Identify the free variables and the bound variables in the following
sentences. Assume, unless indicated otherwise, that the range of every
variable is the set of real numbers.
1
∀x ∈ R, x 2 + 7x ≥ xy
Answer: The variable x is bound because of the quantifier “∀x ∈ R”.
The variable y is free since the is no quantifier for y .
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
3 / 22
example
Example
Identify the free variables and the bound variables in the following
sentences. Assume, unless indicated otherwise, that the range of every
variable is the set of real numbers.
1
∀x ∈ R, x 2 + 7x ≥ xy
Answer: The variable x is bound because of the quantifier “∀x ∈ R”.
The variable y is free since the is no quantifier for y .
2
∃x ∈ R and ∃y ∈ R such that x 2 = y 2 and x > y .
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
3 / 22
example
Example
Identify the free variables and the bound variables in the following
sentences. Assume, unless indicated otherwise, that the range of every
variable is the set of real numbers.
1
∀x ∈ R, x 2 + 7x ≥ xy
Answer: The variable x is bound because of the quantifier “∀x ∈ R”.
The variable y is free since the is no quantifier for y .
2
∃x ∈ R and ∃y ∈ R such that x 2 = y 2 and x > y .
Both x and y are bound because of the quantifiers “∃x ∈ R” and
“∃y ∈ R”.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
3 / 22
Free variables (or parameters)
A sentence in which all variables are bound will, in general, either be a
true sentence or a false sentence.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
4 / 22
Free variables (or parameters)
A sentence in which all variables are bound will, in general, either be a
true sentence or a false sentence.
For example the sentence
∀x ∈ R, ∃y ∈ R such that y 2 + xy + 2 = 0
(1)
contains no free variables and is either a false sentence or a true sentence.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
4 / 22
Free variables (or parameters)
A sentence in which all variables are bound will, in general, either be a
true sentence or a false sentence.
For example the sentence
∀x ∈ R, ∃y ∈ R such that y 2 + xy + 2 = 0
(1)
contains no free variables and is either a false sentence or a true sentence.
On the other hand the sentence
∃y ∈ R such that y 2 + xy + 2 = 0
(2)
has x as a free variable and is neither true nor false.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
4 / 22
Facts about free variables
The following facts give us some idea of the role played by free variables.
(Free variables are also sometimes referred to as parameters).
A free variable, say x, whose domain in the set A represents a fixed
but unspecified element of A.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
5 / 22
Facts about free variables
The following facts give us some idea of the role played by free variables.
(Free variables are also sometimes referred to as parameters).
A free variable, say x, whose domain in the set A represents a fixed
but unspecified element of A.
Usually a sentence P(x) in which the variable x occurs free will be
neither true nor false. Note, however, that if the sentence is altered by
placing a quantifier for x in front or by replacing the free variable x by
some specific element of A then the resulting sentence will be either a
true sentence or a false sentence. For example, equation (1) is the
result of placing a universal quantifier ∀x at the beginning of (2).
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
5 / 22
Facts about free variables
The following facts give us some idea of the role played by free variables.
(Free variables are also sometimes referred to as parameters).
A free variable, say x, whose domain in the set A represents a fixed
but unspecified element of A.
Usually a sentence P(x) in which the variable x occurs free will be
neither true nor false. Note, however, that if the sentence is altered by
placing a quantifier for x in front or by replacing the free variable x by
some specific element of A then the resulting sentence will be either a
true sentence or a false sentence. For example, equation (1) is the
result of placing a universal quantifier ∀x at the beginning of (2).
There are rules which limit the introduction of free variables into a
mathematical argument. These rules will be detailed later.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
5 / 22
The binary connectives
The connective words “and,” “or,” “if . . . then . . .” and “if and only if”
are the binary connectives. Each of these connectives is used to combine
two simpler sentences to form a more complex one. The meanings of these
connective words when they are used in a mathematical setting are very
close to their English meanings. The primary difference is that when the
connectives are used in ordinary English, the meanings may vary depending
on the context whereas in mathematics the meanings are fixed and can be
described by the truth tables.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
6 / 22
The truth table for “and”
Assume that P and Q are (mathematical) statements. Then “P and Q,”
“P or Q,” “if P then Q” and “P if and only if Q” are statements the
truth or falsity of which is entirely determined by the truth or falsity of P
and of Q in a way which can be described by a truth table. For example,
the truth table for “P and Q” is
P
T
T
F
F
Q
T
F
T
F
P and Q
T
F
F
F
Each row of the table gives a truth value for P, a truth value for Q and
the corresponding truth value for “P and Q”. For example, if P is the
sentence “0 < 7” and Q is the sentence “7 ≤ 10” then, since P and Q are
both true, the sentence “P and Q” is true (using the first line of the truth
table.)
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
7 / 22
The truth tables for the other binary connectives are
P
T
T
F
F
Q
T
F
T
F
P or Q
T
T
T
F
P
T
T
F
F
P. Howard
Q
T
F
T
F
P
T
T
F
F
Q
T
F
T
F
If P then Q
T
F
T
T
P if and only if Q
T
F
F
T
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
8 / 22
“or” and “ if - then”
The connective ‘or”
described by the truth table is what is sometime known as the “inclusive
‘or’ ” which might also be written as “one or the other or both.”
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
9 / 22
“or” and “ if - then”
The connective ‘or”
described by the truth table is what is sometime known as the “inclusive
‘or’ ” which might also be written as “one or the other or both.”
If - then statements
A sentence of the form “if P then Q” is called an implication. The
sentence P is called the hypothesis of the implication “if P then Q” and
the sentence Q is called the conclusion. An implication “if P then Q” is
seldom used if it is know whether or not the sentence P is true.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, quantifiers
2016
9 / 22
Implicit Quantifiers
One other important point: There are some circumstances under which
the universal quantifier is omitted from a sentence and must be supplied
by the reader. In other words,
there are sentences P(x) (in which a variable x appears to be free) whose
intended meaning is ∀x, P(x).
In this case the quantifier ∀x, which is not explicitly stated, is called an
implicit quantifier. Implicit quantifiers occur most frequently in “If . . .
then . . .” sentences.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers10 / 22
Example
Is the following sentence true or false?
If x is a real number greater than 0 then x 2 + 1 > 0
P. Howard
(3)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers11 / 22
Example
Is the following sentence true or false?
If x is a real number greater than 0 then x 2 + 1 > 0
(3)
Most people would say this sentence is true. This is because they interpret
the sentence to mean
For all real numbers x if x > 0 then x 2 + 1 > 0
This is the intended meaning of the sentence. The quantifier “for all real
numbers x” is implicit. That is, it has to be supplied by the reader. Most
people without knowing anything about quantifiers would unconsciously
supply the missing ∀x in (3) and say that the sentence is true.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers11 / 22
Implicit Quantifiers, Cont’d
Implicit quantifiers in implications. If P(x) and Q(x) are sentences in
which the variable x occurs free where the domain of x is the set A, then
“If P(x) then Q(x)” means “∀x ∈ A, if P(x) then Q(x).” Similarly “P(x)
if and only if Q(x)” means “∀x ∈ A, P(x) if and only if Q(x).”
One example is provided by 3 above; another is
Theorem
If n2 is an even integer, then n is an even integer.
which has the same meaning as
Theorem
For every integer n, if n2 is even then n is even.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers12 / 22
Similarly
Theorem
n2 is an even integer if and only if n is an even integer.
has the same meaning as
Theorem
For every integer n, n2 is even if and only if n is even.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers13 / 22
Definitions
Implicit quantifiers in definitions. In a definition, after making the
quantifiers described in 12 explicit, any remaining variables that are not
otherwise quantified should be quantified by adding a universal quantifier
whose scope is the entire definition.
For example, consider the sentence “Let f be the function from R to R
defined by f (x) = 7x + 3”. The variable x (with range R) occurs in this
definition and is not otherwise quantified. This means that there is an
implicit universal quantifier ∀x ∈ R and this sentence has the same
meaning as
Let f be the function from R to R defined by ∀x ∈ R,
f (x) = 7x + 3.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers14 / 22
The negation operation
The negation of a sentence P is denoted “not P” or ¬P. The truth table
of ¬P is
P not P
T
F
T
F
The negation of P is the sentence that asserts that the sentence P is false.
The negation of a sentence is seldom obtained by placing the word “not”
in front of it. For example, the negation of “0 < 7” might be written “0 is
not less than 7” or “0 6< 7” or “it is not the case that 0 < 7” or possibly
“0 ≥ 7.”
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers15 / 22
Negation example
For any mathematical sentence P with no free variables, exactly one of the
statements P and “not P” will be true and in many instances we will be
trying to decide which is the case. It is therefore helpful to be able to
write the negation of a sentence in a more readable and positive form. In
the present section we discuss some of the rules for doing this.
We have already seen that a sentence of the form ∀x ∈ S, P(x) is false if
and only if there is some element a of the set S such that P(a) is false.
But this happens if and only if “∃x ∈ S such that not P(x)” is true. This
gives one way of rewriting the negation of ∀x ∈ S, P(x):
“not ∀x ∈ S, P(x)” is equivalent to “∃x ∈ S such that not P(x)”.
For example, the negation of ∀x ∈ R, x 2 > 0 could be written “∃x ∈ R
such that not(x 2 > 0)” or in more positive form “∃x ∈ R such that
x 2 ≤ 0.”
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers16 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
P. Howard
Negation
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
P. Howard
Negation
∃x ∈ S, such that not P(x)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P. Howard
Negation
∃x ∈ S, such that not P(x)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P. Howard
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
P and Q
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P and Q
P. Howard
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
(not P) or (not Q)
If P then (not Q)
If Q then (not P)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P and Q
P. Howard
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
(not P) or (not Q)
If P then (not Q)
If Q then (not P)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P and Q
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
(not P) or (not Q)
If P then (not Q)
If Q then (not P)
P or Q
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P and Q
P or Q
P. Howard
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
(not P) or (not Q)
If P then (not Q)
If Q then (not P)
(not P) and (not Q)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P and Q
P or Q
not P
P. Howard
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
(not P) or (not Q)
If P then (not Q)
If Q then (not P)
(not P) and (not Q)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P and Q
P or Q
not P
P. Howard
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
(not P) or (not Q)
If P then (not Q)
If Q then (not P)
(not P) and (not Q)
P
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P and Q
P or Q
not P
If P then Q
P. Howard
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
(not P) or (not Q)
If P then (not Q)
If Q then (not P)
(not P) and (not Q)
P
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
Quantifiers, connectives and negations
The table below gives a list of sentences involving the connectives that we
have discussed and one or more ways of writing the negation of each.
(Notice that we have used the abbreviation “s.t.” for the words “such
that” in the table.)
Sentence
∀x ∈ S, P(x)
∃x ∈ S such that P(x)
P and Q
P or Q
not P
If P then Q
P. Howard
Negation
∃x ∈ S, such that not P(x)
∀x ∈ S, not P(x)
(not P) or (not Q)
If P then (not Q)
If Q then (not P)
(not P) and (not Q)
P
P and (not Q)
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers17 / 22
If P(x) then Q(x)
(meaning “∀x, If P(x) then Q(x)”
or
“∀x ∈ S, If P(x) then Q(x)”)
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers18 / 22
If P(x) then Q(x)
(meaning “∀x, If P(x) then Q(x)”
or
“∀x ∈ S, If P(x) then Q(x)”)
P. Howard
“∃x s.t. P(x) and (not Q(x))”
or
“∃x ∈ S s.t. P(x) and (not Q(x)”
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers18 / 22
Notes on negations
We usually handle the negation of an “if and only if” statement by
using the fact that “P if and only if Q” is equivalent to “(If P then
Q) and (if Q then P)”.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers19 / 22
Notes on negations
We usually handle the negation of an “if and only if” statement by
using the fact that “P if and only if Q” is equivalent to “(If P then
Q) and (if Q then P)”.
the first two lines of the table tell us that “not ∀x ∈ S, P(x)” is
equivalent to “∃x ∈ S such that not P(x)” and “not ∃x ∈ s such
that P(x)” is equivalent to “∀x ∈ S, not P(x)”. In other words the
“not” can be moved across a quantifier if the quantifier is changed
from ∀ to ∃ or from ∃ to ∀.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers19 / 22
Notes on negations
We usually handle the negation of an “if and only if” statement by
using the fact that “P if and only if Q” is equivalent to “(If P then
Q) and (if Q then P)”.
the first two lines of the table tell us that “not ∀x ∈ S, P(x)” is
equivalent to “∃x ∈ S such that not P(x)” and “not ∃x ∈ s such
that P(x)” is equivalent to “∀x ∈ S, not P(x)”. In other words the
“not” can be moved across a quantifier if the quantifier is changed
from ∀ to ∃ or from ∃ to ∀.
To handle negations of sentences involving more than one connective
negate one connective at a time working from the outside in.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers19 / 22
Negating sentences with more than one connective
For example the negation of “∀x ∈ R, ∃y ∈ R such that y < x and
x < y 2 ” could be simplified in four steps.
¬(∀x ∈ R, ∃y ∈ R such that y < x and x < y 2 ) is equivalent to
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers20 / 22
Negating sentences with more than one connective
For example the negation of “∀x ∈ R, ∃y ∈ R such that y < x and
x < y 2 ” could be simplified in four steps.
¬(∀x ∈ R, ∃y ∈ R such that y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ¬(∃y ∈ R such that y < x and x < y 2 ) is equivalent to
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers20 / 22
Negating sentences with more than one connective
For example the negation of “∀x ∈ R, ∃y ∈ R such that y < x and
x < y 2 ” could be simplified in four steps.
¬(∀x ∈ R, ∃y ∈ R such that y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ¬(∃y ∈ R such that y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ∀y ∈ R, ¬(y < x and x < y 2 ) is equivalent to
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers20 / 22
Negating sentences with more than one connective
For example the negation of “∀x ∈ R, ∃y ∈ R such that y < x and
x < y 2 ” could be simplified in four steps.
¬(∀x ∈ R, ∃y ∈ R such that y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ¬(∃y ∈ R such that y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ∀y ∈ R, ¬(y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ∀y ∈ R, (¬(y < x) or ¬(x < y 2 )) is equivalent to
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers20 / 22
Negating sentences with more than one connective
For example the negation of “∀x ∈ R, ∃y ∈ R such that y < x and
x < y 2 ” could be simplified in four steps.
¬(∀x ∈ R, ∃y ∈ R such that y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ¬(∃y ∈ R such that y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ∀y ∈ R, ¬(y < x and x < y 2 ) is equivalent to
∃x ∈ R such that ∀y ∈ R, (¬(y < x) or ¬(x < y 2 )) is equivalent to
∃x ∈ R such that ∀y ∈ R, (y ≥ x) or (x ≥ y 2 ))
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers20 / 22
Finally, note that to show that a universally quantified statement
“∀x ∈ S, P(x)” is false we show that its negation “∃x ∈ S such that
P(x)” is true. That is, we find an element a ∈ S such that P(a) is false.
This element a is what we have called a counter example for the
universally quantified statement. Hence to show that a statement “If P(x)
then Q(x)” is false (since “If P(x) then Q(x)” means “∀x ∈ S if P(x)
then Q(x)”) we need to find an element a ∈ S for which “If P(a) then
Q(a)” is false. In other words we need to find an element a ∈ S such that
P(a) is true and Q(a) is false. Such an element a is called a counter
example for “If P(x) then Q(x).”
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers21 / 22
Counter example example
Example
Find a counter example for “If x ∈ N then ∃y ∈ N such that y < x.”
A counter example is an element a for which a ∈ N is true and “∃y ∈ N
such that y < a” is false. a = 0 is such a counter example. Note that for
the sentence of this example there is no other counter example.
P. Howard
The Language of Mathematics - free variables, connectives
September
and implicit
12, 2016
quantifiers22 / 22