Mathematical Logic and Set Notations

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Unit 2

• Mathematical Logic and Set Notations

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Part A Mathematical Logic

• 2.1 Simple Propositions, Conjunction, Disjunction
  and Negation of Propositions

A simple proposition is a simple statement which
has a definite meaning and it is either true or
false, but not both. The truth value of a true
proposition is true, denoted by T, while that
of a false proposition is false, denoted by F.
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Part A Mathematical Logic
2.1 Simple Propositions, Conjunction, Disjunction
and Negation of Propositions
-ve e.g.

• Can you solve the problem?
• Lets go!
• Get a pencil for me.
• He is the student.
• x 2 8

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Part A Mathematical Logic
2.1 Simple Propositions, Conjunction, Disjunction
and Negation of Propositions
Composite proposition is one which is formed by
connecting several simple propositions, called
components, with connectives.
e.g. (1) John is a boy and Mary is a girl.
(2) Snow is white or the sun rises from the
West. (3) If today is Friday then the
earth is spherical.
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Part A Mathematical Logic
2.1 Simple Propositions, Conjunction, Disjunction
and Negation of Propositions
Let P and Q be two simple propositions. The
composite propositions P and Q, denoted by P ? Q,
is called the conjunction of P and Q. The truth
value P ? Q is denoted by the following truth
table.
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Part A Mathematical Logic
2.1 Simple Propositions, Conjunction, Disjunction
and Negation of Propositions
Let P and Q be two simple propositions. The
composite propositions P and Q, denoted by P v Q,
is called the disjunction of P and Q. The truth
value P v Q is denoted by the following truth
table.
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Part A Mathematical Logic
2.1 Simple Propositions, Conjunction, Disjunction
and Negation of Propositions
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Part A Mathematical Logic
2. Equivalence of Two Propositions
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Part A Mathematical Logic
2.2 Equivalence of Two Propositions
De Morgans Law
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Part A Mathematical Logic
2.2 Equivalence of Two Propositions
De Morgans Law
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Part A Mathematical Logic
2.2 Equivalence of Two Propositions
De Morgans Law
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P.50 Ex. 2A
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Part A Mathematical Logic
2.3 Conditional Propositions
Let P and Q be two propositions, the composite
proposition, denoted by P?Q and read if P then
Q, whose truth value is defined by the truth
table
is called a conditional proposition in which P is
called the antecedent and Q is the consequence.
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Part A Mathematical Logic
2.3 Conditional Propositions
e.g. Let P denotes a triangle is isosceles., Q
denotes the base angles of a triangle are
equal. P?Q is the proposition If a triangle is
isosceles then the base angles of the triangle
are equal.
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Part A Mathematical Logic
2.3 Conditional Propositions
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Part A Mathematical Logic
2.3 Conditional Propositions
Let P?Q be a conditional proposition. Then P is
called the sufficient condition for Q and Q is
the necessary condition for P.
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Part A Mathematical Logic
2.3 Conditional Propositions
e.g. If I study hard, I shall have a happy
life. The proposition can be interpreted that
studying hard guarantees me having a happy life.
Studying hard is a sufficient condition for
having a happy life. Having a happy life is the
necessary condition for studying hard.
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Part A Mathematical Logic
3. Conditional Propositions

• Let P?Q be a conditional proposition. This
  proposition has the following three derivatives
• the converse Q?P,
• the inverse (P)?(Q),
• the contrapositive (Q)?(P)

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Part A Mathematical Logic
2.3 Conditional Propositions
Consider the following proposition If x is an
even integer, then x is divisible by 2.

• It has three derivatives
• the converse If x is divisible by 2, then x is
  an even integer.
• the inverse If x is not an even integer, then
  x is not divisible by 2.
• the contrapositive If x is not divisible by 2,
  then x is not an even integer.

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Part A Mathematical Logic
2.3 Conditional Propositions
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Part A Mathematical Logic
2.3 Conditional Propositions
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Part A Mathematical Logic
2.3 Conditional Propositions
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Part A Mathematical Logic
2.3 Conditional Propositions
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Part A Mathematical Logic
2.3 Conditional Propositions
Proof by contrapositive.
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Theorem 1
Given tangent PQ, point of contact T
Suppose PQ is not perpendicular to OT. We can
1) draw a perpendicular OR ? PQ
and 2) take a point S on PQ such that SR RT.
Then ? ORT ? ? ORS (SAS)
? OS OT radius of the circle
? S is a point on the circle.
? PQ cuts the circle at points T and S.
This is false because PQ is a tangent.
? The assumption is incorrect.
? PQ ? OT
tangent ? radius
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Part A Mathematical Logic
2.3 Conditional Propositions
Proof by contradiction.
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Part A Mathematical Logic
2.3 Conditional Propositions
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Part A Mathematical Logic
2.3 Conditional Propositions
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Part A Mathematical Logic
2.3 Conditional Propositions
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Part A Mathematical Logic
2.4 Biconditional Propositions
Let P and Q be two propositions. The
biconditional proposition
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Part A Mathematical Logic
2.4 Biconditional Propositions
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Part A Mathematical Logic
2.4 Biconditional Propositions

• Note
• iff is an abbreviation for if and only if.
• The biconditional proposition P Q can be read
  as
• (a) P if and only if.
• (b) P when and only when Q.
• (c) The necessary and sufficient conditions
  for P is Q.
• (d) The necessary and sufficient conditions
  for Q is P.

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Part A Mathematical Logic
2.4 Biconditional Propositions
(1) True conditional and biconditional
propositions are transitive.
(2) True biconditional proposition is reflexive.
(i.e. a conditional proposition is equivalent to
its contrapositive.)
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
gt
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
gt
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
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Part A Mathematical Logic
2.5 Universal and Existential Quantifiers
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P.61 Ex.2B
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Part B Set Notations
2.6 Concept and Notations of Sets
A collection of objects is called a set. The
object of a set is called the element.
There are two ways of expressing a set.
(1) Tabular form List all the elements of the
set and put them inside a parenthesis.
e.g. 1, 2, 3, 4 is a set containing four
elements.
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Part B Set Notations
2.6 Concept and Notations of Sets
There are two ways of expressing a set.
(2) Set-builder form Put down the form and the
properties of all the elements of the set inside
a parenthesis.
e.g. x x is an even number is a set
containing all the even numbers. (x, y) x is
the father of y is a set containing all the
pairs (x, y) such that x and y possess father-son
relationship.
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Part B Set Notations
2.6 Concept and Notations of Sets
Note (1) In set-builder form, the two dots are
sometimes written as a stroke, i.e. x x is an
even number can be written as x x is an even
number
(2) We accept the notation of a set as an
undefined concept, such as points and lines in
plane geometry.
(3) Usually, we use capital letters A, B, C,. to
denote sets and use small letters a, b, c,to
denote the elements of a set.
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Part B Set Notations
2.6 Concept and Notations of Sets
For x ? A, x is an element of A, x is a member of
A, x is contained in A or A contains x.
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Part B Set Notations
2.6 Concept and Notations of Sets
Sets containing finite number of elements are
called finite sets. Otherwise, they are called
infinite sets.
finite set e.g. x x is an integer and 1lt xlt 5
infinite set e.g. x x is an integer
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Part B Set Notations
2.6 Concept and Notations of Sets
A set containing no element is called an empty
set, denoted by ?.
e.g. x x is a real number and sin x 2
-ve e.g. ? is not an empty set.
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Part B Set Notations
2.6 Concept and Notations of Sets
A set containing exactly one element is called a
singleton.
e.g. 2 is a singleton.
e.g. 1, 1, 1is a singleton.
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Part B Set Notations
2.6 Concept and Notations of Sets
Symbols frequently use
(1) N denotes the set of all positive integers
(or natural numbers)
(2) Z denotes the set of all integers.
(3) Q denotes the set of rational numbers.
(4) Q denotes the set of all positive rational
numbers.
(5) R denotes the set of all real numbers.
(6) R denotes the set of all positive real
numbers.
(7) C denotes the set of all complex numbers.
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Part B Set Notations
2.6 Concept and Notations of Sets
Venn diagrams are used to illustrate sets and
relations between sets.
e.g. A 1, 2, 3, 4 B 3, 4, 5, 6,
7, 8 C 7, 8 D 9, 10
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Part B Set Notations
2.7 Subsets
Let A and B be two sets. B is a subset of A if
and only if every element of B is an element of A.
e.g. A 1, 2, 3, 4 B 3, 4,
B is a subset of A. B is included in A. A
includes B. i.e.
B is not a subset of A. i.e.
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Part B Set Notations
2.7 Subsets
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Part B Set Notations
2.7 Subsets
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Part B Set Notations
2.7 Subsets
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Part B Set Notations
2.7 Subsets
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Part B Set Notations
2.7 Subsets
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Part B Set Notations
2.7 Subsets
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Part B Set Notations
2.7 Subsets
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Part B Set Notations
2.8 Simple Operations of Sets
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Part B Set Notations
2.8 Simple Operations of Sets
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Part B Set Notations
2.8 Simple Operations of Sets
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Part B Set Notations
2.8 Simple Operations of Sets
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Part B Set Notations
2.8 Simple Operations of Sets
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Part B Set Notations
2.8 Simple Operations of Sets
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Part B Set Notations
2.8 Simple Operations of Sets
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Part B Set Notations
2.8 Simple Operations of Sets
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P.68 Ex.2C
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Part B Set Notations
2.9 Recognition of Relations and Functions
If x, y y, x, the objects x and y appear is
immaterial to the construction of the set x, y.
The set x, y is called the unordered pair.
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Part B Set Notations
2.9 Recognition of Relations and Functions
e.g. Let A 1, 2, 3, and B 4, 5
A x B (1, 4), (1, 5), (2, 4), (2, 5), (3, 4),
(3, 5)
B x A (4, 1), (4, 2), (4, 3), (5, 1), (5, 2),
(5, 3)
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Part B Set Notations
2.9 Recognition of Relations and Functions
(2) Geometrically, the set R is the real number
line and R x R ( or R2) is the Cartesian plane.
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Part B Set Notations
2.9 Recognition of Relations and Functions
e.g. A a, b, c, B 1, 2 R
(a,1), (a,2), (b,2) S (b,1), (c,2)
R and S are relations from A to B.
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
e.g. A 1, 2, 3, 4, 5, B w, x,
y, z and R (1, w), (1, x),(2, x),
(3, y)
Pr1R 1, 2, 3 and Pr2R w, x, y
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
f A ? B
A is the domain, B is the range and Pr2f is the
image of the function f.
A Dom( f ), B Rang( f ), Pr2f Im( f )
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
Let f A?B be a function from A to B. For each
element a of A, we denote by f(a) the unique
element of B such that (a, f(a))? f. f(a) is
called the value of f at a or the image of a
under f, a is called the pre-image of f(a) under
f. We also say that f maps a into (or onto) f(a).
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Part B Set Notations
2.9 Recognition of Relations and Functions
f1(1) a, f1(2) b, f1(3) b, f1(4)
c f2(1) d, f2(2) c, f2(3) b, f2(4) a.
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Part B Set Notations
2.9 Recognition of Relations and Functions
A function of real variable is a function whose
domain is the set of all real numbers R or a
subset of R. A real-valued function is a function
whose range is the set of all real numbers R or a
subset of R. A function defined on a set A is a
function whose domain is the set A.
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
9. Recognition of Relations and Functions
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Part B Set Notations
9. Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
1
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
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Part B Set Notations
2.9 Recognition of Relations and Functions
It is clear that the inverse function of a
bijection is also a bijection.
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Part B Set Notations
2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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2.9 Recognition of Relations and Functions
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P.76 Ex.2D
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Part B Set Notations
2.10 Proof of injective and Surjective Functions
To prove a function (mapping) f is injective, we
always use the technique Suppose f(a) f(b)
and then to show a b
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Part B Set Notations
2.10 Proof of injective and Surjective Functions
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2.10 Proof of injective and Surjective Functions
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Part B Set Notations
2.10 Proof of injective and Surjective Functions
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Part B Set Notations
2.10 Proof of injective and Surjective Functions
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P.80 Ex.2E