Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION

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MTH001 Elementary Mathematics

LECTURE # 1

1.Recommended Books:

1.Discrete Mathematics with Applications (second edition) by Susanna S. Epp

MAIN TOPICS:

1. Logic

2. Sets & Operations on sets

3. Relations & Their Properties

4. Functions

5. Sequences & Series

Set of Integers:

-2

-1

Set of Real Numbers:

-3

-2

-1

What is Discrete Mathematics?:

Discrete Mathematics concerns processes that consist of a sequence of individual steps.

LOGIC:

Logic is the study of the principles and methods that distinguishes between a

valid and an invalid argument.

SIMPLE STATEMENT:

A statement is a declarative sentence that is either true or false but not both.

A statement is also referred to as a proposition

Example: 2+2 = 4, It is Sunday today

If a proposition is true, we say that it has a truth value of "true".

If a proposition is false, its truth value is "false".

The truth values "true" and "false" are, respectively, denoted by the letters T and F.

EXAMPLES:

Not Propisitions

1.Grass is green.

Close the door.

2.4 + 2 = 6

2.4 + 2 = 7

x is greater than 2.

3.There are four fingers in a hand.

He is very rich

are propositions

are not propositions.

Rule:

If the sentence is preceded by other sentences that make the pronoun or variable reference

clear, then the sentence is a statement.

Example

Example:

Bill Gates is an American

x=1

He is very rich

x>2

He is very rich is a statement with truth-value

x > 2 is a statement with truth-value

TRUE.

FALSE.

UNDERSTANDING STATEMENTS:

1.x + 2 is positive.

Not a statement

2.May I come in?

Not a statement

3.Logic is interesting.

A statement

4.It is hot today.

A statement

5.-1 > 0

A statement

6.x + y = 12

Not a statement

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MTH001 Elementary Mathematics

COMPOUND STATEMENT:

Simple statements could be used to build a compound statement.

EXAMPLES:

LOGICAL CONNECTIVES

1. "3 + 2 = 5" and "Lahore is a city in Pakistan"

2. "The grass is green" or " It is hot today"

3. "Discrete Mathematics is not difficult to me"

AND, OR, NOT are called LOGICAL CONNECTIVES.

SYMBOLIC REPRESENTATION:

Statements are symbolically represented by letters such as p, q, r,...

EXAMPLES:

p = "Islamabad is the capital of Pakistan"

q = "17 is divisible by 3"

CONNECTIVE

MEANINGS

SYMBOL

CALLED

Negation

not

Tilde

∧

Conjunction

and

Hat

∨

Disjunction

Vel

→

Conditional

if...then...

Arrow

↔

Biconditional

if and only if

Double arrow

EXAMPLES:

p = "Islamabad is the capital of Pakistan"

q = "17 is divisible by 3"

p ∧ q = "Islamabad is the capital of Pakistan and 17 is divisible by 3"

p ∨ q = "Islamabad is the capital of Pakistan or 17 is divisible by 3"

~p = "It is not the case that Islamabad is the capital of Pakistan" or simply

"Islamabad is not the capital of Pakistan"

TRANSLATING FROM ENGLISH TO SYMBOLS:

Let p = "It is hot", and q = " It is sunny"

SENTENCE

SYMBOLIC FORM

1.It is not hot.

p ∧q

2.It is hot and sunny.

p∨q

3.It is hot or sunny.

~ p ∧q

4.It is not hot but sunny.

~p∧~q

5.It is neither hot nor sunny.

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MTH001 Elementary Mathematics

EXAMPLE:

Let

h = "Zia is healthy"

w = "Zia is wealthy"

s = "Zia is wise"

Translate the compound statements to symbolic form:

(h ∧ w) ∧ (~s)

1.Zia is healthy and wealthy but not wise.

~w ∧ (h ∧ s)

2.Zia is not wealthy but he is healthy and wise.

~h ∧ ~w ∧ ~s

3.Zia is neither healthy, wealthy nor wise.

TRANSLATING FROM SYMBOLS TO ENGLISH:

Let

m = "Ali is good in Mathematics"

c = "Ali is a Computer Science student"

Translate the following statement forms into plain English:

1.~ c

Ali is not a Computer Science student

2.c ∨ m

Ali is a Computer Science student or good in Maths.

3.m ∧ ~c

Ali is good in Maths but not a Computer Science student

A convenient method for analyzing a compound statement is to make a truth

table for it.

A truth table specifies the truth value of a compound proposition for all

possible truth values of its constituent propositions.

NEGATION (~):

If p is a statement variable, then negation of p, "not p", is denoted as "~p"

It has opposite truth value from p i.e.,

if p is true, ~p is false; if p is false, ~p is true.

TRUTH TABLE FOR

CONJUNCTION (∧):

If p and q are statements, then the conjunction of p and q is "p and q", denoted as

"p ∧ q".

It is true when, and only when, both p and q are true. If either p or q is false, or

if both are false, p∧q is false.

TRUTH TABLE FOR

p∧q

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MTH001 Elementary Mathematics

DISJUNCTION (∨)

or INCLUSIVE OR

If p & q are statements, then the disjunction of p and q is "p or q", denoted as

"p ∨ q".It is true when at least one of p or q is true and is false only when both

p and q are false.

TRUTH TABLE FOR

p∨q

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MTH001 Elementary Mathematics

Note that in the table T is only in that row where both p and q have T

and all other values are F. Thus for finding out the truth values for the conjunction of two

statements we will only first search out where the

both statements are true and write down the T in the corresponding row

in the column of p ∧ q and in all other rows we will write F in the

column of p ∧ q.

DISJUNCTION (∨) or INCLUSIVE OR

If p and q are statements, then the disjunction of p and q is "p or q", denoted as "p ∨ q"

It is true when at least one of p or q is true and is false only when both p and q are false.

Note that in the table F is only in that row where both p and q have F and all other values

are T. Thus for finding out the truth values for the disjunction of two statements we will only

first search out where the both statements are false and write down the F in the

corresponding row in the column of p ∨ q and in all other rows we will write T in the column

of p ∨ q.

Remark:

Note that for Conjunction of two statements we find the T in both the

statements, But in disjunction we find F in both the statements. In other words we will

fill T first in the column of conjunction and F in the column of disjunction.

SUMMARY

1. What is a statement?

2. How a compound statement is formed.

3. Logical connectives (negation, conjunction, disjunction).

4. How to construct a truth table for a statement form.

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