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” …..
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.
- 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
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
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”…..
p=It is below freezing.
q=It is snowing.
p^q=It is below freezing and it is snowing.
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”.
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
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.
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.
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
“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.
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.
Truth table for p ↔q
Precedence of logical operators
In compound proposition order of operators is applied..