Discrete Mathematics

Defecation of Discrete Mathematics:

Concerns processes that consist of a sequence of individual step are called Discrete Mathematics

Continuous Discrete

LOGIC:

Logic study of principles and method that distinguishes/difference b/w valid and invalid argument is

called logic.

Simple statement:

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

A statement is also referred to as a proposition

Examples:

1. If a proposition is true then we say that a value is true.

2. And if proposition is false then we say that truth value is false.

3. Truth & false are denoted by T and F

Examples:

Propositions: Not Proposition:

1. Grass is green. Close the door.

2. 4+3=6. X is greater than 2.

3. 4+2=6. He is very rich.

Rules:

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

the sentence is a statement.

Examples:

False True

I. X=1 Bill gates is an American

II. x>3 He is very Rich

III. x>3 is a statement with truth-value He is very rich is a statement with

truth-value

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

Compound Statement:

Simple statement could be used to build a compound statement.

Examples:

1. 3+2=5 and Multan is a city of Pakistan

2. The grass is green or it is hot today

3. Ali is not very rich

And, Not, OR are called Logical Connectives.

SYMBOLIC REPRESENTATION:

Statement is symbolically represented by letters such as “p, q, r…

EXAMPLES:

p=”Multan is a city of Pakistan”

q=”17 is divisible by 3”

CONNECTIV MEANING SYMBOL CALLED NEGATION NOT ˜ TILDE

CONJUNCTION AND ^ HAT DISJUNCTION OR v VEL

CONDITIONAL IF……THEN ARROW

BICONDITIONAL IF AND ONLY IF DOUBLE ARROW

EXAMPLES:

p=”Multan is a city of Pakistan”

q=”Ali is a Muslim”

p ^ q=”Multan is a city of Pakistan” AND “Ali is a Muslim”

p v q=” Multan is a city of Pakistan” OR” Ali is a Muslim”

˜p=”Multan is not a city of Pakistan”

TRANSLATING FROM ENGLISH TO SYMBOLIS:

Let p=”it is cold”, AND “it is Ali”

SENTENCE SYMBOLIC

1. It is not cold ˜P

2. It is cold AND Ali P ^ q

3. It is cold OR Ali p v q

4. It is NOT cold BUT Ali ˜p ^ q

COMPOUND STATEMENT EXAMPLES:

Let a=”Ali is Healthy” b=”Ali is Wealthy” c=”Ali is Wise”

i. Ali is healthy AND wealthy But NOT wise. (a ^ b) ̂ (˜c)

ii. Ali is NOT healthy BUT he is wealthy AND wise. (˜a) ̂ (b ̂ c)

iii. Ali is NEITHER healthy, Wealthy NOR wise. (˜a ^ ~ b v ~ c)

TRANSLATING FROM SYMBOLS TO ENGLISH:

Let: m=”Ali is a good in math” c=”Ali is a com. science student”

I. ~ C Ali is “NOT” com. Science student.

II. C v m Ali is com. Science student” OR “good in math.

III. M ^ ~ c Ali is good in math” BUT AND NOT “a com. Science student.

WHAT IS TRUTH TABLE?

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

constituent proposition.

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

NEGATION (~)

If p=statement variable, then negation of p “NOT p”, is denoted by “~p”

If p is true, ~p is false

If p is false ~p is true

TRUTH TABLE FOR ~P

P ~P

T F

F T

CONJUNCTION (^)

If p and q is statement then conjunction is “p and q”

Denoted by “p ^q”

If p and q are true then true

If both or either false then False

P ^q

P q P ^q T T T

T F F F T F

F F F

DISJUNCTION (v)

If P and q is statement then “p or q”

Denoted by “p v q”

If both are false then false

If both or either is true then true

P v q

P q P v q T T T

T F T

F T T F F F

Truth Table for this statement ~p^ q

P q ~p ~p ^q T T F F

T F F F

F T T T F F T F

Truth Table for ~p^ (q v ~r)

P q r ~r (q v ~r) ~p ~p ^(q v ~r) T T T F T F F

T T F T T F F

T F T F F F F T F F T T F F

F T T F T T T F T F T T T T

F F T F F T F F F F T T T T

Truth table for (p v q) ̂ ~ (p ̂ q)

P q (p v q) (p ^ q) ~(p ^q) (p v q)^~(p ^q) T T T T F F

T F T F T T

F T T F T T F F F F T F

Double negation property ~ (~p) =p

P (~p) ~(~p) T F T

F T F

So it is clear that “p” and double negation of “p” is equal.

Example

English to symbolic

P= I am Umair Shah

~p= I am not Umair Shah

~ (~p) = I am Umair Shah

So it is clear that double negation of “p” is also equal to “p”.

~ (p ̂ q) & ~p ^~q are not Equal.

P q (p ^q) ~(p ^q) ~p ~q ~p ^~q T T T F F F F

T F F T F T F F T F T T F F

F F F T T T T

So it is clear that “~ (p ̂ q) & ~p ^~q” are not equal

De Morgan’s Law

1. The negation of “AND” statement is logically equivalent to the “OR” statement in which each

component is negated.

Symbolically ~ (p ^q) = ~p v ~q

P q P ^q ~(p ^q) ~p ~q ~p v ~q

T T T F F F F T F F T F T T

F T F T T F T F F F T T T T

So it is clear that ~ (p ̂ q) = ~p v ~q are logically equivalent

2. The negation of “OR” statement is logically equivalent to the “AND” statement in which component

is negated.

Symbolically ~ (p v q) = ~p ^ ~q

P q (P v q) ~(p v q) ~p ~q ~p ^ ~q

T T T F F F F T F T F F T F

F T T F T F F F F F T T T T

So it is clear that ~ (p v q) = ~p ^ ~q is Equal.

Application

Negation for each of the following:

a. The fan is slow or it is very hot

b. Ali is fit or Awais is injured.

Solution:

a. The fan is not slow “AND” it is not very hot

b. Ali is not fit “AND” Awais is not injured

Inequalities and DE MORGANES law:

Exercise:

1. (p ^q) ^ r = P^(q ^ r)

p q r (p ^q) (p ^q)^r (q ^r) P^(q ^r) T T T T T T T

T T F T F F F T F T F F F F

T F F F F F F F T T F F T F

F T F F F F F

F F T F F F F F F F F F F F

So it clears that Colum 5 and Colum 7 are equal

2. (P ^q) v r = p ^ (q v r)?????

P q r (p ^q) (p ^q) v r (q v r) P ^(q v r) T T T T T T T

T T F T T T T

T F T F T T T T F F F F F F

F T T F T T F F T F F F T F

F F T F T T F F F F F F F F

So it clear that Colum 5 and Colum 7 are not equal.