Mathematics in the Modern World - Quantifiers

OBJECTIVES
TITLE
LOGIC
PROPOSITION
A.
CONJUNCTION
LOGICAL STATEMENT
and
QUANTIFIERS
Reporter: Arreza, Daniela Jane B.
Maglinte, ReyJan C.

EXCLUSIVE-OR
TABLE 9.1
B.
TABLE 9.2
C.
EXAMPLE
PROPORTIONAL
FUNCTION
EXISTENTIAL
QUANTIFIER
UNIVERSAL
QUANTIFIER
TABLE 9.3
QUANTIFIERS
ARREZA, DANIELA JANE B
BSED 1 C

EXCLUSIVE-OR
TABLE 9.1
B.
TABLE 9.2
C.
EXAMPLE
PROPORTIONAL
FUNCTION
EXISTENTIAL
QUANTIFIER
UNIVERSAL
QUANTIFIER
TABLE 9.3
Objectives
At the end of the lesson, students are
expected to:
• Describe quantifiers;
• Identify the two types of quantifiers; and
• Apply the two types of quantifiers.

EXCLUSIVE-OR
TABLE 9.1
B.
TABLE 9.2
C.
EXAMPLE
QUANTIFIERS
• Predicate – a predicate (or open statement) is a
statement whose truth depends on the value of
one or more variables.
• It becomes propositions once every variable is
bound by assigning a universe of discourse.
• It can also be denoted by a function-like notation.
Example:
P(x) = “x is an even number.”
So now, P(2) is true, and P(3) is false. If P is
predicate, then P(x) is either true or false, depending
on the value of x.
PROPORTIONAL
FUNCTION
EXISTENTIAL
QUANTIFIER
UNIVERSAL
QUANTIFIER
TABLE 9.3

EXCLUSIVE-OR
TABLE 9.1
B.
TABLE 9.2
C.
EXAMPLE
PROPORTIONAL
FUNCTION
PROPORTIONAL FUNCTION:
• It is a sentence P(x); it becomes a
statement only when variable x is given a
particular value.
• It denoted as P(x), Q(x), R(x), and so on.
Example:
“If x is an odd number, then x is not a multiple
of 2.”
So it’s logical form is P(x) → Q(x).
EXISTENTIAL
QUANTIFIER
UNIVERSAL
QUANTIFIER
TABLE 9.3
TABLE 9.4 & 9.5
END

TABLE 9.2
C.
EXAMPLE
EXISTENTIAL
QUANTIFIER
PROPORTIONAL
FUNCTION
UNIVERSAL
QUANTIFIER
UNIVERSAL QUANTIFIER:
• The universal quantifier
is symbolized by ∀.
• The statement “for all x,
P(x),” is symbolized by
∀𝑥 𝑃(𝑥).
• “∀𝑥 𝑃(𝑥)” is true if only if
P(x) is true for every
value of x.
TABLE 9.3
TABLE 9.4 & 9.5
END

TABLE 9.2
C.
EXAMPLE
EXISTENTIAL
QUANTIFIER
PROPORTIONAL
FUNCTION
UNIVERSAL
QUANTIFIER
Example 1:
TABLE 9.3
TABLE 9.4 & 9.5
END
Let P(x) be a statement x + 1 > x
P(x) is true for all positive integers x
Or
Ɐx P(x)
For all
Quantifier
P(1) : 1 + 1 > 1 = 2 > 1 (TRUE)
P (2) : 2 + 1 > 2 = 3 > 2 (TRUE)

TABLE 9.1
B.
TABLE 9.2
C.
EXAMPLE
EXISTENTIAL
QUANTIFIER
PROPORTIONAL
FUNCTION
EXISTENTIAL QUANTIFIER:
• Its symbol is ∃.
• The statement “there exists
an x such for which P(x),” is
symbolized by ∃𝑥 𝑃(𝑥).
• “∃𝑥 𝑃(𝑥)” is only true if there
is at least one value of x for
which P(x) is true.
UNIVERSAL
QUANTIFIER
TABLE 9.3
TABLE 9.4 & 9.5
END

TABLE 9.1
B.
TABLE 9.2
C.
EXAMPLE
EXISTENTIAL
QUANTIFIER
PROPORTIONAL
FUNCTION
Example 2:
UNIVERSAL
QUANTIFIER
TABLE 9.3
TABLE 9.4 & 9.5
END
Let Q(x) be the statement x < 2
Q(1) : 1 < 2 (TRUE)
Q(2) : 2 < 2 (FALSE)
Q: Is there some value of x for which Q(x) is True?
(If domain is set of positive integers)
A: Yes. Q(x) is true for x = 1.
Therefore, there is an x for which Q(x) is True
or
ⱻx Q(x)
There exist
Quantifier

C.
EXAMPLE
EXISTENTIAL
QUANTIFIER
PROPORTIONAL
FUNCTION
UNIVERSAL
QUANTIFIER
TABLE 9.3
If the universe of discourse for P is
𝑃{𝑝1, 𝑝2, … , 𝑝𝑛}, then ∀𝑥 𝑃 𝑥 ↔ 𝑃 𝑝1 ∧ 𝑃 𝑝2 ∧ ⋯ ∧
𝑃(𝑝𝑛) and ∃𝑥 𝑃 𝑥 ↔ 𝑃 𝑝1, ∨ 𝑃 𝑝2 ∨ ⋯ ∨ 𝑃 𝑝𝑛 .
From this, we can easily determine the truth
values of the quantifiers.
TABLE 9.4 & 9.5
END

C.
EXISTENTIAL
QUANTIFIER
PROPORTIONAL
FUNCTION
UNIVERSAL
QUANTIFIER
TABLE 9.3
TABLE 9.4 & 9.5
END

EXISTENTIAL
QUANTIFIER
PROPORTIONAL
FUNCTION
UNIVERSAL
QUANTIFIER
TABLE 9.3
TABLE 9.4 & 9.5
END
THANK
YOU

Mathematics in the Modern World - Quantifiers

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