MMW_Chap_2_Mathematical_Language_and_Symbols.ppt.pptx

Learning
Objectives
o Discuss the language, symbols and conventions of
mathematics.
o Explain the nature of mathematics as a language.
o Acknowledge that mathematics is a useful language.
o Compare and contrast expression and sentences.
o Identify and discuss the four basic concepts in
mathematical language.
o List and discuss some basic operations on logic and
logical formalities.
o Perform operations on mathematical expressions
correctly.
o Articulate the importance of mathematics in one’s life.
o Express appreciation for mathematics as a human
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.

Topic
Outline
I. Characteristics of Mathematical
Language
II. Expression versus Sentences
III. Conventions in the Mathematical
Language
IV. Four Basic Concepts
V. Elementary Logic
VI. Formality

Conventions in the Mathematical
Language
Mathematics is a spoken and written natural languages for
expressing mathematical language.
Mathematical language is an efficient and powerful tool for
mathematical expression, exploration, reconstruction after
exploration, and communication.
It is precise and concise.
It is has a poor understanding of the language.

Language
Mathematics languages:
Digits 0, 1, 2, 3, 4, 5, 6, 7,
8, 9;
Mathematical symbols

Mathematical
Language
Mathematical language is the system used to communicate
mathematical ideas.
It consists of some natural language using technical terms
(mathematical terms) and grammatical conventions that are
uncommon to mathematical discourse, supplemented by a
highly specialized symbolic notation for mathematical
formulas.
Mathematical notation used for formulas has its own
grammar and shared by mathematicians anywhere in the
globe.
Mathematical language is being precise, concise, and
powerful.

Expression versus
Sentences
An expression (or mathematical expression) is a finite
combination of symbols that is well-defined according to
rules that depend on the context.
Symbols can designate numbers, variables, operations,
functions, brackets, punctuations, and groupings to help
determine order of operations, and other aspects of
mathematical syntax.
Expression – correct arrangement of mathematical symbols
to represent the object of interest, does not contain a
complete thought, and cannot be determined if it is true or
false.
Some types of expressions are numbers, sets, and functions.

Expression versus
Sentences
Sentence (or mathematical sentence) – a statement about
two expressions, either using numbers, variables, or a
combination of both.
Uses symbols or words like equals, greater than, or less than.
It is a correct arrangement of mathematical symbols that
states a complete thought and can be determined
whether it’s true, false, sometimes true/sometimes false.

Language
Sentence (or mathematical sentence) – a statement about
two expressions, either using numbers, variables, or a
combination of both.
Uses symbols or words like equals, greater than, or less than.
It is a correct arrangement of mathematical symbols that
states a complete thought and can be determined
whether it’s true, false, sometimes true/sometimes false.

Language
Different and specific meaning within mathematics—
group ring field term factor,
Special
terms—
tensor fractal
functor
Mathematical Taxonomy —
Axiom conjecture
theorem
s
lemm
a
corollaries

Language
= (equal)
+ (addition)
(division)
< (less-
than)
– (subtraction)
(element)
> (greater-than)
(multiplication)
(for all)
(implies)
(there exists) (infinity)
(if and only if) (approximately) (therefore)
Formulas are written predominantly left to right, even when
the writing system of the substrate language is right-to-left.
Latin alphabet is commonly used for simple variables
and parameters.
Mathematical expressions

FourBasic
Concepts
A. Language of Sets
B. Language of Functions
C.Language of Relations
D. Language of Binary
Operations

Language of
Sets
Set theory is the branch of mathematics that studies sets or
the mathematical science of the infinite.
George Cantor (1845-1918) is a
German Mathematician
He is considered as the founder of
set theory as a mathematical
discipline.

Sets and
Elements
A set is a well-defined collection of objects.
The objects are called the elements or members of the
set.
element of a set
not an element of a set.

Some Examples of
Sets
A = {x xis a positive integer less than 10}
B = {x xis a real number and x2 – 1 =
0} C = {x xisa letter in the word dirt}
D = {x xis an integer, 1 x 8}
E = {x xis a set of vowel
letters}
Set E equals the set of all x such that x is a set of
vowel letters” or E = {a, e, i, o, u}

Indicate whether the ff. defined a
Set
a. The list of course offerings of Centro Escolar
University.
Answer: Set
b. The elected district councilors of Manila City.
Answer: Set
c. The collection of intelligent monkeys in Manila
Zoo. Answer: Not a set

List the Elements of the
Sets
a. A = {x xis a letter in the word
mathematics.} Answer: A = {m, a, t, h, e, i,
c, s.}
b. B = {x xis a positive integer, 3 x 8.}
Answer: B = {3, 4, 5, 6, 7, 8}
c. C = {x x= 2n + 3, n is a positive
integer.} Answer:C = {5, 7, 9, 11, 13,
…}

Methods of Writing
Sets
Roster Method. The elements of the set are enumerated
and separated by a comma it is also called tabulation
method.
Rule Method. A descriptive phrase is used to describe
the elements or members of the set it is also called set
builder notation, symbol it is written as {x P(x)}.
Example:
E = {a, e, i, o, u} Roster method
E = {x x is a collection of vowel letters}Rule method

Write the ff. Sets in Roster
Form
a. A= {x xis the letter of the word discrete}
Answer: A = {d, i, s, c, r, e, t}
b. B = {x 3 x 8, x Z}
Answer: B = {4, 5, 6,
7}
c. C = {x xis the set of zodiac signs}
Answer: C = {Aries, Cancer, Capricorn, Sagittarius,
Libra, Leo, …}

Write the ff. Sets using Rule
Method
a. D = {Narra, Mohagany, Molave, …}
Answer: D = {x xis the set of non-bearing trees.}
b. E = {DOJ, DOH, DOST, DSWD, DENR, CHED,
DepEd,…} Answer: E = {x xis the set of
government agencies.}
c. F = {Botany, Biology, Chemistry, Physics, …}
Answer: F = {x xis the set of science
subjects.}

Some Terms on
Sets
Finite and Infinite
Sets.
Unit Set
Empty Set
Universal Set
Cardinality

Finite
Set
Finite set is a set whose elements are limited or countable,
and the last element can be identified.
Example:
a. A = {x xis a positive integer less than 10}
b.C = {d, i, r, t}
c.E = {a, e, i, o, u}

Infinite
Set
Infinite set is a set whose elements are unlimited or
uncountable, and the last element cannot be specified.
Example:
a. F = {…, –2, –1, 0, 1, 2,…}
b. G = {x xis a set of whole numbers}
c. H = {x xis a set of molecules on earth}

Unit
Set
A unit set is a set with only one element it is also
called singleton.
Example:
a.I = {x xis a whole number greater than 1 but less than 3}
b.J = {w}
c.K = {rat}

Empty
Set
An empty set is a unique set with no elements (or null set), it
is denoted by the symbol or { }.
Example:
a.L = {x xis an integer less than 2 but greater than 1}
b.M = {x xis a number of panda bear in Manila Zoo}
c.N = {x xis the set of positive integers less than zero}

Universal
Set
Universal set is the all sets under investigation in any
application of set theory are assumed to be contained in
some large fixed set, denoted by the symbol U.
Example:
a. U = {x xis a positive integer, x2 = 4}
b. U = {1, 2, 3,…,100}
c. U = {x xis an animal in Manila Zoo}

Cardinalit
y
The cardinal number of a set is the number of elements or
members in the set, the cardinality of set A is denoted by n(A)
Example: Determine its cardinality of the ff. setsAnswer
a. E = {a, e, i, o, u}, n(E) =
5
b. A = {x x is a positive integer less than 10} n(A) = 9
c. C = {d, i, r, t} A = {1, 2, 3, 4, 5, 6, 7, 8, 9} n(C) = 4
Theorem 1.1: Uniqueness of the Empty Set: There is only one
set with no elements.

Venn
Diagram
Venn Diagram is a pictorial presentation of relation and opera
on set.
Also known set diagrams, it show all hypothetically possible
logical relations between finite collections of sets.
Constructed with a collection of simple
closed curves drawn in the plane or
normally comprise of overlapping circl
The interior of the circle symbolically
represents the elements (or members)
the set, while the exterior represents
elements which are not members of th
set.
Introduced by John Venn in his paper "On the Diagrammatic

Kinds of
Sets
Subset
Proper
Subset
Equal Set
Power Set

Subs
et
If A and B are sets, A is called subset of B, if and only if,
every element of A is also an element of B.
Symbolically: A B x,x A x B.
Example:
Suppose A = {c,
d, e}
B = {a, b, c, d, e}
U = {a, b, c, d, e, f, g}
Then A B, since all elements of A is in B.

Proper
Subset
Let A and B be sets. A is a proper subset of B, if and only if,
every element of A is in B but there is at least one element
of B that is not in A.
The symbol denotes that it is not a proper
subset. Symbolically: A B x,x A x B.
Example:
Suppose A = {c,
d, e}
B = {a, b, c, d, e}
C = {e, a, c, b, d}
U = {a, b, c, d, e, f, g}
Then A B, since all elements of A is in B.

Equal
Sets
Given set A and B, A equals B, written, if and only if,
every element of A is in B and every element of B is in
A.
Symbolically: A = B A B B A.
Example:
Suppose A = {a, b, c, d, e},
B = {a, b, d, e, c}
U = {a, b, c, d, e, f, g}
Then then A B and B A, thus A = B.

Power
Set
Given a set S from universe U, the power set of S
denoted by
(S), is the collection (or sets) of all subsets of S.
Example
:
Determine the power set of(a) A = {e, f},
(b) = B = {1, 2,
3}.
(a) A = {e,
f}
(A) = {{e}, {f}, {e, f}, }
(b) B = {1, 2,
3}
(B) = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2,
3},
{1, 2, 3}, }.

Theore
m
Theorem 1.2: A Set with No Elements is a Subset of Every
Set: If
is a set with no elements and A is any set,
then
A.
Theorem 1.3: For all sets A and B, if A B then (A) (B).
Theorem 1.4: Power Sets: For all integers n, if a set S
has n elements then (S) has 2n elements.

Operations on
Sets
Union
Intersection
Complement
Difference
Symmetric
Difference
Disjoint Sets
Ordered Pairs

Unio
n
The union of A and B, denoted A B, is the set of all
elements x in U such that x is in A or x is in B.
Symbolically: A B= {x x A x B}.

Intersectio
n
The intersection of A and B, denoted A B, is the set of
all elements x in U such that x is in A and x is in B.
Symbolically: A B= {x x A x B}.

Compleme
nt
The complement of A (or absolute complement of A),
denoted A’,is the set of all elements x in U such that x is
not inA.
Symbolically: A’ = {x U x A}.

Differenc
e
The difference of A and B (or relative complement of B
with respect to A), denoted A B, is the set of all
elements x in U such that x is in A and x is not in B.
Symbolically: A B = {x x A x B} = A B’.

Symmetric
Difference
If set A and B are two sets, their symmetric difference as the
set consisting of all elements that belong to A or to B, but not
to both A and B.
Symbolically: A B = {x x (A B) x (A B)}
= (A B) (A B)’or (A B) (A B).

Exampl
e
Suppose
A = {a, b,
c}
B = {c, d,
e}
U = {a, b, c, d, e, f,
g}
Find the following
a. A B
b. A B
c. A’
d. A B
e. A B

Solutio
n
a. A B = {a, b, c, d,
e}
b. A B = {c}
c.A’ = {d, e, f, g}
d. A = {a,
b} B
e. A
B
= {a, b, d,
e}

Disjoint
Sets
Two set are called disjoint (or non-intersecting) if and only
if, they have no elements in common.
Symbolically: A and B are disjoint A B= .

Order
Pairs
In the ordered pair (a, b), a is called the first component and
b is called the second component. In general, (a, b) (b, a).
Example: Determine whether each statement is true or false.
a. (2, 5) = (9 – 7, 2 + 3) SiTnrcue2= 9 – 7 and 2 + 3 = 5, the
ordered pair
is equal.
b. {2, 5} {5, 2}
c. (2, 5) (5, 2)
SiFnaclesetheseare sets and not ordered
pairs, the order in which the elements are
listed is not important.
T
T
h
r
e
u
s
eordered pairs are not equal since
they do not satisfy the requirements for
equality of ordered pairs.

Cartesian
Product
The Cartesian product of sets A and B, written AxB,
is AxB = {(a, b) a A and b B}
Example: Let A = {2, 3, 5} and B = {7, 8}. Find each set.
a. AxB = {(2, 7), (2, 8), (3, 7), (3, 8), (5, 7), (5, 8)}
b. BxA = {(7, 2), (7, 3), (7, 5), (8, 2), (8, 3), (8, 5)}
c. AxA = {(2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5), (5, 2), (5,
3),
(5, 5)}

Language of Functions and
Relations
A relation is a set of ordered pairs.
If x and y are elements of these sets and if a relation exists
between x and y, then we say that x corresponds to y or that
y depends on x and is represented as the ordered pair of (x,
y).
A relation from set A to set B is defined to be any subset of
A B.
If R is a relation from A to B and (a, b) R, then we say that “a
is related to b” and it is denoted as a R b.

Relations
Let A = {a, b, c, d} be the set of car brands, and
B = {s, t, u, v} be the set of countries of the car
manufacturer.
Then A Bgives all possible pairings of the elements of A and
B, let the relation R from A to B be given by
R = {(a, s), (a, t), (a, u), (a, v), (b, s), (b, t), (b, u), (b, v), (c, s),
(c, t), (c, u), (b, v), (d, s), (d, t), (d, u), (d, v)}.

Relations
Let R be a relation from set A to the set B.
domain of R is the set dom R
dom R = {a A (a, b) R for some b B}.
image (or range) of R
im R = {b B (a, b) R for some a A}.
Example: A = {4, 7},
Then A A= {(4, 4), (4, 7), (7, 4),(7, 7)}.
Let on A be the description of x y x + y is
even. Then (4, 4) ,and (7, 7) .

Relations
Function is a special kind of relation helps visualize
relationships in terms of graphs and make it easier to
interpret different behavior of variables..
Applications of
Functions:
financial applications
Engineering
calculating pH levels
designing
machineries
economics medicine
sciences natural
disasters measuring
decibels

Language of Binary
Operations
A function is a relation in which, for
each value of the first component of
the ordered pairs, there is exactly
one value of the second
component.
The set X is called the domain of
the function.
For each element of x in X, the corresponding element y in
Y is called the value of the function at x, or the image of x.
Range – set of all images of the elements of the domain is
called the of the function. A function can map from one set
to another.

Language of Binary
Operations
Determine whether each of the following relations is a
function.
A = {(1, 3), (2, 4), (3, 5), (4, 6)}
B = {(–2, 7), (–1, 3), (0, 1), (1, 5), (2, 5)}
C = {(3, 0), (3, 2), (7, 4), (9, 1)}

Language of Binary
Operations
Algebraic structures focuses on investigating sets associated
by single operations that satisfy certain reasonable axioms.
An operation on a set generalized structures as the
integers together with the single operation of addition, or
invertible 2 2matrices together with the single operation
of matrix multiplication.
The algebraic structures known as group.

Binary
Operations
Let G be a set. A binary operation on G is a function that
assigns each ordered pair of element of G.
Symbolically, a b = G, for all a, b, c G.

Grou
p
A group is a set of elements, with one operation, that
satisfies the following properties:
(i)the set is closed with respect to the operation,
(ii)the operation satisfies the associative
property, (iii)there is an identity element, and
(iv)each element has an inverse.

Grou
p
A group is an ordered pair (G, )where G is a set and
is a binary operation on G satisfying the four
properties.
Closure property. If any two elements are
combined using the operation, the result must be an
element of the set. a b = c G, for all a, b, c G.
Associative property. (a b) c = a (b c), for all a,
b,
c G.
Identity property. There exists an element e in G,
such
that for all a G, a e = e a.
Inverse property. For each a G there is an element
a–1
of G, such that a a–1 = a–1 a = e.

Grou
p
The set of group G contain all the elements including the
binary operation result and satisfying all the four
properties closure, associative, identity e, and inverse a–
1.

Exampl
e
Determine whether the set of all non-negative integers
under addition is a group.
Solution:
Apply the four properties to test the set of all non-negative
integersunderadditionisagroup.
Step 1: Closure property, choose any two positive
integers, 8 + 4 = 12 and 5 + 10 = 15
The sum of two numbers of the set, the result is
always a number of the set.
Thus, it is closed.

Solutio
n
Step 2: Associative property, choose three positive
integers
3 + (2 + 4) = 3 + 6 = 9
(3 + 2) + 4 = 5 + 4 = 9
Thus, it also satisfies the associative property.
Step 3: Identity property, choose any positive
integer 8 + 0 = 8; 9 + 0 = 9; 15 + 0 =
15
Thus, it also satisfies the identity property.

Solutio
n
Step 4: Inverse property, choose any positive integer
4 + (–4) = 0;
10 + (–10) = 0;
23 + (–23) = 0
Note that a–1 = –a.
Thus, it also satisfies the inverse property.
Thus, the set of all non-negative integers under addition
is a group, since it satisfies the four properties.

Formal
Logic
The science or study of how to evaluate arguments &
reasoning.
It differentiate correct reasoning from poor reasoning.
It is important in sense that it helps us to reason
correctly.
The methods of reasoning.

Mathematical
Logic
Mathematical logic (or symbolic logic) is a branch
of mathematics with close connections to
computer science.
Mathematical study of logic and the applications of formal
logic to other areas of mathematics.
It also study the deductive formal proofs systems and
expressive formal systems.
Four Divisions:
Set
Theory
Proof Theory Model
Theory
Recursion
Theory

Aristotle (382-322
BC)
Aristotle is generally regarded
as the Father of Logic
The study started in the late 19th
century with the development of
axiomatic frameworks for analysis,
geometry and arithmetic.

Stateme
nt
A statement (or proposition) is a
declarative sentence which is either true or
false, but not both.
The truth value of the statements is the truth and falsity of
the statement.

Exampl
e
Which of the following are
statements?
1. Manila is the capital of the
Philippines.
Is true
A
statement.
2. What day is
it?
It is a
question Not
a statement.
It cannot be categorized as true or
false. Not a statement.
3. Help me,
please.
4. He is
handsome.
Is neither true nor false - “he” is not
specified.
Not a
statement.

Ambiguous
Statements
1. Mathematics is fun.
2. Calculus is more interesting than
Trigonometry.
3. It was hot in Manila.
4. Street vendors are poor.

Propositional
Variable
A variable which used to represent a statement.
A formal propositional written using propositional logic
notation, p, q, and r are used to represent statements.

Logical
Connectives
Logical connectives are used to combine simple statements
which are referred as compound statements.
A compound statement is a statement composed of two or
more simple statements connected by logical connectives
A statement which is not compound is said to be simple
(also called atomic).
“or” “not”
“and” “exclusive-
or.”
“if then”
“if and only if”

Conjunctio
n
The conjunction of the statement p and q is the
compound statement “p and q.”
Symbolically, p q, where is the symbol for “and.”
Property 1: If p is true and q is true, then p q is true;
otherwise p q is false. Meaning, the conjunction of
two statements is true only if each statement is
true.
p q p q
T T T
T F F
F T F
F F F

Exampl
e
Determine the truth value of each of the following
conjunction.
3. Ferdinand Marcos is the only three-term Philippine
PresideTntrue and Joseph Estrada is the only Philippine
President who resigns.
1. 2 + 6 = 9 and man is a
mammal. False True
2. Manny Pacquiao is a boxing champion and Gloria
Macapagal Arroyo is the first female Philippine
President.
Fals
e
Fals
e

Stateme
nt
The disjunction of the statement p, q is the compound
statement “p or q.”
Symbolically, p q, where is the symbol for “or.”
Property 2: If p is true or q is true or if both p and q are true,
then p q is true; otherwise p q is false. Meaning,
the disjunction of two statements is false only if
each statement is false.
p q p q
T T T
T F T
F T T
F F F

Exampl
e
Determine the truth value of each of the following
disjunction.
1.2 + 6 = 9 or Manny Pacquiao is a boxing champion. True
False True
2. Joseph Ejercito is the only Philippine President who
resignsTorrue Gloria Macapagal Arroyo is the first female
Philippine President.
3. Ferdinand Marcos is the only three-term Philippine
True President or man is a mammal.

Negatio
n
The negation of the statement p is denoted by p,where
is the symbol for “not.”
Property 3: If p is true, p is false. Meaning, the truth value
of the negation of a statement is always the
reverse of the truth value of the original
statement.
p p
T
F
F
T

Exampl
e
The following are statements for p, find the
corresponding p.
1. 3 + 5 =
8.
2. Sofia is a
girl.
3. Achaiah is not
here.
3 + 5 8.
Sofia is a
boy.
Achaiah is
here.

Condition
al
The conditional (or implication) of the statement p and q is the
compound statement “if p then q.”
Symbolically, p q, where is the symbol for “if then.” p is called
hypothesis (or antecedent or premise) and q is called conclusion
(or consequent or consequence).
Property 4: The conditional statement p
q is false only when p is true and q is
false; otherwise p q is true. Meaning p
q states that a true statement cannot
imply a false statement.
p q p q
T T T
T F F
F T T
F F T

Exampl
e
In the statement “If vinegar is sweet, then sugar is
sour.”
The antecedent is “vinegar is sweet,”
and the consequent is
“sugar is sour.”

Exampl
e
Obtain the truth value of each of the following
conditional statements.
1. If vinegar is sweet, then sugar is sour.
2. 2 + 5 = 7 is a sufficient condition for 5 + 6
= 1.
True
Fals
e
True
Fals
e
Fals
e
Tru
e
Fals
e
3. 14 – 8 = 4 is a necessary condition that 6 3
= 2. False True

Bicondition
al
The biconditional of the statement p and q is the
compound statement “p if and only if q.”
Symbolically, p q, where is the symbol for “if and only if.”
Property 5: If p and q are true or both false, then p q is
true; if p and q have opposite truth values, then p
q is false.
p q p q
T T T
T F F
F T F
F F T

Exampl
e
Determine the truth values of each of the following
biconditional statements.
1. 2 + 8 = 10 if and only if 6 – 3 = 3. True
2. Manila is the capital of the Philippines is
equivalent to fish live in moon.
3. 8 – 2 = 5 is a necessary and sufficient for 4 +
2 = 7.
Fals
e
True
Tru
e
Tru
e

Exclusive-
Or
The exclusive-orof the statementp and q is the
compound statement “p exclusive or q.”
Symbolically, p q, where is the symbol for “exclusive or.”
Property 6: If p and q are true or both false, then p q is
false; if p and q have opposite truth values, then p
q is true.
p q p q
T T F
T F T
F T T
F F F

Exampl
e
“Sofia will take her lunch in Batangas or she will
have it in Singapore.”
Case 1: Sofia cannot have her lunch in Batangas
and at the same time in do it in Singapore,”
Case 2: If Sofia will have her lunch in Batangas
or in Singapore, meaning she can only
have it in one location given a single
schedule.
Case 3: If she ought to decide to have her lunch
elsewhere (neither in Batangas nor in
Singapore).
Fals
e
True
Fals
e

Predicat
e
A predicate (or open statements) is a statement whose truth
depends on the value of one or more variables.
Predicates become propositions once every variable is
bound by assigning a universe of discourse.
Most of the propositions are define in terms of predicates
Example:
“x is an even number” is a predicate whose truth
depends on the value of x.

Predicat
e
A predicate can also be denoted by a function-like notation.
Example:
P(x) = “x is an even number.” Now P(2) is true, and
P(3) is false.
If P is a predicate, then P(x) is either true or false, depending
on the value of x.

Propositional
Function
A propositional function is a sentence P(x); it becomes a
statement only when variable x is given particular value.
Propositional functions are denoted as P(x), Q(x),R(x), and
so on.
The independent variable of propositional function must
have a universe of discourse, which is a set from which the
variable can take values.

Propositional
Function
Example:
“If x is an odd number, then x is not a multiple of 2.”
The given sentence has the logical form P(x) Q(x) and its
truth value can be determine for a specific value of x.
Exampl
e:
Existential
Quantifiers
There exists an x such that x is odd number and 2x is even
number.
For all x, if x is a positive integer, then 2x + 1 is an odd number.
Universal Quantifiers

Universe of
Discourse
The universe of discourse for the variable x is the set of
positive real numbers for the proposition
“There existsan x suchthat x is odd numberand 2x is
even number.”
Binding variable is used on the variable x, we can say
that the occurrence of this variable is bound.
A variable is said to be free, if an occurrence of a variable
is not bound.

Universe of
Discourse
To convert a propositional function into a proposition, all
variables in a proposition must be bound or a particular
value must be designated to them.
This is done by applying combination of quantifiers
(universal, existential) and value assignments.
The scope of a quantifier is the part of an assertion in which
variables are bound by the quantifier.
A variable is free if it is outside the scope of all quantifiers.

Existential
Quantifiers
The statement “there exists an x such that P(x),” is
symbolized by xP(x).
The symbol is called the existential quantifier
The statement “ xP(x)”is true if there is at least one value of
x for which P(x) is true.

Universal
Quantifiers
The statement “for all x, P(x),” is symbolized by xP(x).
The symbol is called the universal quantifier.
The statement “ xP(x)”is true if only if P(x) is true for
every value of x.

Topic
Outline
Quantifier Symbol Translation
Existential There exists
There is
some For
some
For which
For at least
one Such that
Satisfying
Universal For all
For each
For
every
For any
Given
any

Truth Values of
Quantifiers
If the universe of discourse for P is P{p1, p2, …, pn},
then
xP(x) P(p1) P(p2) …P(pn) and
xP(x) P(p1) P(p2) …P(pn).
Statement Is True when Is False when
xP(x) P(x) is true for
every x.
There is at
least one x for
which P(x) is
false.
xP(x) There is at
least one x for
which P(x) is
true.
P(x) is false
for every x.

QuantifiedStatements andtheir
Negation
Statement Negation
All A are B. Some A are not B.
No A are B. Some A are B.
Some A are not B. All A are B.
Some A are B. No A are B.

For the things of this world
cannot be made known
without a knowledge of
mathematics.
– Roger
Bacon
Copyright 2018:
Mathematics in the Modern World by Winston S.

MMW_Chap_2_Mathematical_Language_and_Symbols.ppt.pptx

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