CHAPTER 2.pdf

Mathematical
Language and
Symbols
Chapter 2:
GEC 1
Prepared by:
Polane S. Gumiran

Topic Outline
Copyright 2018: Mathematics in the Modern World by Winston S. Sirug, Ph.D.
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.

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.

Mathematical Convention is a fact, name, notation, or usage
which is generally agreed upon by mathematicians.
PEMDAS (Parenthesis, Exponent, Multiplication, Division,
Addition and Subtraction.)
All mathematical names and symbols are conventional.

Different and specific meaning within mathematics—
group ring field term factor,
Special terms—
tensor fractal function
Mathematical Taxonomy —
Axiom conjecture theorems lemma corollaries

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

Four Basic 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
D = {xx is an integer, 1  x  8}
A = {xx is a positive integer less than 10}
B = {xx is a real number and x2 – 1 = 0}
C = {xxis a letter in the word dirt}
E = {xx is 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.
b. The elected district councilors of Manila City.
c. The collection of intelligent monkeys in Manila Zoo.
Answer:
Answer:
Answer:
Set
Set
Not a set

List the Elements of the Sets
a. A = {xx is a letter in the word mathematics.}
b. B = {xx is a positive integer, 3  x  8.}
c. C = {xx = 2n + 3, n is a positive integer.}
Answer:
Answer:
Answer:
A = {m, a, t, h, e, i, c, s.}
B = {3, 4, 5, 6, 7, 8}
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}
E = {xx is a collection of vowel letters}
Roster method
Rule method

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

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

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

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

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

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

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

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:
c. U = {xx is an animal in Manila Zoo}
b. U = {1, 2, 3,…,100}
a. U = {xx is a positive integer, x2 = 4}

Cardinality
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. sets
c. C = {d, i, r, t}
b. A = {xx is a positive integer less than 10}
a. E = {a, e, i, o, u}, n(E) = 5
n(A) = 9
n(C) = 4
Answer
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
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 operations
on set.
Also known set diagrams, it show all hypothetically possible logical
relations between finite collections of sets.
Introduced by John Venn in his paper "On the Diagrammatic and
Mechanical Representation of Propositions and Reasoning’s"
Constructed with a collection of simple closed
curves drawn in the plane or normally
comprise of overlapping circles.
The interior of the circle symbolically
represents the elements (or members) of the
set, while the exterior represents elements
which are not members of the set.

Kinds of Sets
 Subset
 Proper Subset
 Equal Set
 Power Set

Subset
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.
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.
C = {e, a, c, b, d}
The symbol  denotes that it is not a proper subset.

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}, }.

Theorem
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

Union
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}.

Intersection
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}.

Complement
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 in A.
Symbolically: A’ = {x  U  x  A}.

Difference
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).

Example
Suppose
B = {c, d, e} U = {a, b, c, d, e, f, g}
A = {a, b, c}
Find the following
a. AB
b. AB
c. A’
d. A  B
e. A  B

Solution
a. AB
b. AB
d. A  B
e. A  B
= {a, b, c, d, e}
= {c}
= {a, b}
= {a, b, d, e}
c. A’ = {d, e, f, g}

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)
b. {2, 5}  {5, 2}
c. (2, 5)  (5, 2)
Since 2 = 9 – 7 and 2 + 3 = 5, the ordered pair
is equal. True
Since these are sets and not ordered pairs,
the order in which the elements are listed is
not important. False
These ordered pairs are not equal since they
do not satisfy the requirements for equality
of ordered pairs. True

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

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B gives 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)}.

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)  .

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 economics medicine
Engineering sciences natural disasters
calculating pH levels measuring decibels
designing machineries

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.

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)}

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2
matrices 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.

Group
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.

Group
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.

Group
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.

Example
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
integers under addition is a group.
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.

Solution
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.

Solution
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.
Four Divisions:
Model Theory
Set Theory Recursion Theory
Proof Theory
 It also study the deductive formal proofs systems and
expressive formal systems.
 Mathematical study of logic and the applications of formal
logic to other areas of mathematics.

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.

Statement
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.

Example
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.
3. Help me, please. It cannot be categorized as true or false.
Not a statement.
4. He is handsome. Is neither true nor false - “he” is not
specified.
Not a statement.

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

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

Conjunction
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
F
F
T
F
T
F
T
F
F
F

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

Statement
The disjunction of the statement p, q is the compound
statement “p or q.”
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.
Symbolically, p  q, where  is the symbol for “or.”
p q p  q
T
T
F
F
T
F
T
F
T
T
T
F

Example
Determine the truth value of each of the following disjunction.
1. 2 + 6 = 9 or Manny Pacquiao is a boxing champion.
2. Joseph Ejercito is the only Philippine President who
resigns or Gloria Macapagal Arroyo is the first female
Philippine President.
3. Ferdinand Marcos is the only three-term Philippine
President or man is a mammal.
False True
True
True
True

Negation
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

Example
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.

Conditional
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
F
F
T
F
T
F
T
F
T
T

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

Example
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.
3. 14 – 8 = 4 is a necessary condition that 6  3 = 2.
True
False
True
False False
True False
False True

Biconditional
The biconditional of the statement p and q is the compound
statement “p if and only if q.”
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.
Symbolically, p  q, where  is the symbol for “if and only if.”
p q p  q
T
T
F
F
T
F
T
F
T
F
F
T

Example
Determine the truth values of each of the following
biconditional statements.
1. 2 + 8 = 10 if and only if 6 – 3 = 3.
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.
True
False
True
True True

Exclusive-Or
The exclusive-or of the statement p and q is the compound
statement “p exclusive or q.”
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.
Symbolically, p  q, where  is the symbol for “exclusive or.”
p q p  q
T
T
F
F
T
F
T
F
F
T
T
F

Example
“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).
False
True
False

Predicate
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
“x is an even number” is a predicate whose truth depends
on the value of x.
Example:

Predicate
A predicate can also be denoted by a function-like notation.
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.
Example:

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
“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.
There exists an x such that x is odd number and 2x is even number.
Example:
For all x, if x is a positive integer, then 2x + 1 is an odd
number.
Example: Existential Quantifiers
Universal Quantifiers

Universe of Discourse
The universe of discourse for the variable x is the set of
positive real numbers for the proposition
“There exists an x such that x is odd number and 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.
The scope of a quantifier is the part of an assertion in which
variables are bound by the quantifier.
This is done by applying combination of quantifiers
(universal, existential) and value assignments.
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 x P(x).
The symbol  is called the existential quantifier
The statement “x P(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 x P(x).
The symbol  is called the universal quantifier.
The statement “x P(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
x P(x)  P(p1)  P(p2) … P(pn) and
x P(x)  P(p1)  P(p2) … P(pn).
Statement Is True when Is False when
x P(x) P(x) is true for
every x.
There is at least
one x for which
P(x) is false.
x P(x) There is at least
one x for which
P(x) is true.
P(x) is false for
every x.

Quantified Statements and their 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.

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