What is Discrete Mathematics?
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:
1.Grass is green.
2.4 + 2 = 6
2.4 + 2 = 7
3.There are four fingers in a hand.
are propositions
Not Propositions
• Close the door.
• x is greater than 2.
• He is very rich
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:
x = 1
x > 2
x > 2 is a statement with truth-value
FALSE.
Example
Bill Gates is an American
He is very rich
He is very rich is a statement with truth-value
TRUE.
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
COMPOUND STATEMENT:
Simple statements could be used to build a compound statement.
EXAMPLES:
1.“3 + 2 = 5” and “Delhi is a city in India”
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 = “Delhi is the capital of India”
q = “17 is divisible by 3”
EXAMPLES:
p = “Delhi is the capital of India”
q = “17 is divisible by 3”
p ∧ q = “Delhi is the capital of India and 17 is divisible by 3”
p ∨ q = “Delhi is the capital of India or 17 is divisible by 3”
~p = “It is not the case that Delhi is the capital of India” or simply
“Delhi is not the capital of India”
TRANSLATING FROM ENGLISH TO SYMBOLS:
Let p = “It is hot”, and q = “ It is sunny”
SENTENCE
1.It is not hot.
2.It is hot and sunny.
3.It is hot or sunny.
4.It is not hot but sunny.
5.It is neither hot nor sunny.
SYMBOLIC FORM
~ p
p ∧q
p ∨ q
~ p ∧q
~ p ∧ ~ q
EXAMPLE:
Let h = “Zia is healthy”
w = “Zia is wealthy”
s = “Zia is wise”
Translate the compound statements to symbolic form:
1.Zia is healthy and wealthy but not wise. (h ∧ w) ∧ (~s)
2.Zia is not wealthy but he is healthy and wise. ~w ∧ (h ∧ s)
3.Zia is neither healthy, wealthy nor wise. ~h ∧ ~w ∧ ~s
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
~ p
~ p
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
p ∧ q
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
p ∨ q
