Before starting to study this course..I would like to suggest how you can best learn **discrete mathematics** you will learn the most by actively working exercise. I suggest you solve exercise as many as possibly can. I encourage you to solve the additional exercises….

In logic and proofs you learn about rules of logic which specify the meaning of mathematical statements.

“**LOGIC**” is the basis of all mathematical reasoning. In mathematics the term “**PROOF**” is that we must know and understand what makes up a correct mathematical argument…If once we prove a mathematical statement is true we call it “**THEOREM**” …..

“**PROPOSITION**”

A proposition is a declarative sentence that is either true or false, but not both..e.g.

- Islamabad is the capital of Pakistan.. Truth value is “
**True**” - 2+3=6……………Truth value is “
**False**”

Some sentences are not proposition like that.

- X+3=5…
- Is today cloudy?

Propositional variables are used to represent propositions, which are p,q,r…….etc. these are used just like that..

- P=”Today is Sunday”
- q=”It is rainy”

The area of logic that deals with proposition is called the” proposition calculus” or “proposition logic”.

## What is Negation in Discrete Mathematics

__Negation__

It is unary operator because it apply on single value..If P be a proposition then **negation of P** denoted by **” ****¬****P** ”. It is read as “**not p**”.If negation operator is apply on single existing proposition new proposition is obtained…This reverse the original value.

Let P=”Today is Friday” the negation of “P” is written as “Today is not Friday”.

__Truth Table for Negation__

Now at this stage we will introduce the logical operators that are used to form new propositions from two or more existing propositions..These logical operators are also called **connectives.**

__Conjunction in Discrete Mathematics__

__Conjunction in Discrete Mathematics__

It is binary operator..Let p and q proposition the conjunction of p and q denoted by p^q and read as “p AND q”.

The conjunction of p and q is **“true” **when both are true otherwise **“false”…..**

**e.g.**

**p**=It is below freezing.

**q**=It is snowing.

**p****^****q=**It is below freezing and it is snowing.

** **

****Note.**.

In logic the world “**but**” is used instead of “**AND**”.

__Truth Table for Conjunction__

**Disjunction in Discrete Mathematics**

Let p and q be proposition the disjunction of p and q is denoted by p˅q this is read as “p OR q”.The value of disjunction is “**False**” when both p and q is false otherwise “**True**”.

**e.g.**

p=The election is decided..

q=The votes have been counted.

p˅q=The election is decided or the votes have been counted.

**Truth Table of disjunction in Discrete Mathematics**

__Exclusive OR __

Let p and q be the proposition. The exclusive or between p and q is denoted by **p****⊕****q **.The value of exclusive OR is true when one of the value p or q is exactly **“True”** otherwise **“False”.**

__Truth Table for Exclusive OR__

**Implication in Discrete Mathematics**

A conditional statement is called an **Implication.**

The statement **p****→****q** is called conditional statement because **p****→****q** assert that q is true when the condition that p holds is true.

****Note.**

The statement **p****→****q** is true when both p and q are true and when q is false the value is “**False**”.

Conditional statement play an important role in mathematical reasoning, a variety of terminology is used to express **p****→****q ** just like that;

“if p then q”

“p implies q”

“q is necessary for p” etc.

**e.g.**

p=I bought a lottery ticket this week.

q=I won the million dollar jackpot on Friday.

**p****→****q**= If I bought a lottery ticket this week, then I won the million dollar jackpot on Friday.

__Truth Table for p____→____q__

__Biconditional in Discrete Mathematics__

__Biconditional in Discrete Mathematics__

**“p if and only if q”.**It is denoted by p ↔q.

Let p and q be the propositions then the biconditional statement **p ****↔q **is true when p and q have the same truth value is it true or false otherwise it is false.Biconditional statements are also called **bi.implication.**

e.g.

p=You can take the flight.

q=you buy a ticket.

**p ****↔q**=you can take flight if and only if you buy a ticket.

__ __

__Also Check What is Discrete Mathematics __

__Truth table for p ____↔q__

__Precedence of logical operators__

In compound proposition order of operators is applied..

**¬****^****˅****→****↔**

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