- Negation
- Conjunction
- Disjunction
- Implications
- Reductions
- Equivalences
→ Truth value:-
The truth and falsity of a statement is called its truth value. The truth value of a statement is denoted by T or F according as it is true or false.
→ Negation:(not)
Let p be any preposition. The negative of given proposition P denoted by ⁓p is called negation of p.
Example:
p-“I love animals.”
negation of p i.e ⁓p is “I do not love animals.”
p: The summer in Terai is very hot.
⁓p: The summer in Terai is not very hot.
Truth Table
| p | ⁓p |
| T | F |
| F | T |
“p” represents the truth value of proposition.
“⁓p” represents the negation of p.
→ Conjunction:(AND)
Given two propositions p and q, the proposition “p and q” denoted ny pΛq is the proposition that is true whenever both the propositions p and q are true, false otherwise.
• The proposition that is obtained by the use of “and” operator is also called conjunction of p and q.
Example:
If we have propositions p=”Ram is smart.” and q=”Ram is intelligent.”
then conjunction of p and q is (pΛq): Ram is smart and intelligent.
• This proposition is true only when Ram is smart and intelligent also, false otherwise.
Truth Table
| p | q | pΛq |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
“p and q” represents the truth value of proposition.
“pΛq” represents the conjunction (AND) operation between p and q.
→ Disjunction:(OR)
Given two propositions p and q, the proposition “p OR q” is denoted by p⋁q is the proposition that is false whenever both the propositions p and q are false, true otherwise.
• The proposition that is obtained by the use of “or” operator is also called disjunction of p and q.
Example:
if we have proposition p= Ram is intelligent and q= Ram is diligent.
then disjunction of p and q is (p⋁q): Ram is intelligent or diligent.
Truth Table
| p | q | p⋁q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
“p and q” represents the truth value of proposition.
“p⋁q” represents the disjunction (OR) operation between p and q.
→ Exclusive OR:(XOR)
Given two propositions p and q, the proposition exclusive or of p and q is denoted by p⨁q is the proposition that is true whenever only onen of the propositions p and q is true, false otherwise.
Example:
if we have propositins p= Ram drinks coffee in the morning. and q= Ram drinks tea in the morning. then the exclusive or of p and q (p⨁q): Ram drinks coffee or tea in the morning.
Truth Table
| p | q | p⨁q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
“p and q” represents the truth value of proposition.
“p⨁q” represents the exclusive or (XOR) operation between p and q.
→ Implication(→):
Given two propositions p and q, the implication p→q is the proposition that is false when p is true and q is false, true otherwise.
•Here p is called hypothesis and q is called consequence.
We use different terminologies to express p→q like:
• if p , then q
• q is the consequence of p
• p only if q
Example:
if we have propositions p=Today is sunday. and q=It is hot today. then implication of p and q (p→q ): If today is sunday then it is hot day.
or Today is sunday only if it is hot today.
Truth Table
| p | q | p→q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
• “p” and “q” represent the truth values of the hypothesis and conclusion.
• “p→q” represents the truth value of the implication.
