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# Proposition

## Proposition

Definition: A proposition is a declarative sentence that is either true or false.
Example: 1. It rained yesterday.
2. Dhaka is the capital of Bangladesh.
A sense which has no true or false answer or value is not proposition.
Example: 1. What time is it?
2. Where are you going?
Tautology: A proposition that is true under all circumstances is referred to as tautology.
Example: (i) 6 is divisible by 3.
(ii) 2 and 2 makes four.
Contradiction: A proposition that is false under all circumstances is referred to as contradiction.
Example: (i) 6 is divisible by 5.
(ii) 2 and 2 makes 5.
Equivalents: Two proposition p and q are said to equivalent if when p is true, q also true and when p is false, q is also false.
Example: Let p denote proposition “ He was born in January /2000.”
and q denote the proposition “He will be years old in December/11.”
Then p and q are said to be equivalent.
Disjunction: Let p and q be two propositions, we define disjunction of p and q denoted by p∨q (∨=join) to be a proposition which is true when either one or both of p and q are true and false when both of p and q are false.
Conjunction: Let p and q be two propositions, we define conjunction of p and q denoted by p∧q (∧=meet) to be a proposition which is true when both of p and q are true and is false when either one or both p and q are false.
Negation: Let P be a proposition, we define the negation of p denoted by p ̅    be a proposition which is true when p is false and is false when p is true.
Atomic proposition: A proposition which is not a combination of other proposition is referred to as atomic proposition.
If p then q: Let p and q be two propositions, we define a proposition “ if p then q” denoted by p→q which is true if both p and q are true or false or if p is false and is false if p is true and q is false.

p                             q                                   p→q

T                             T                                   T

F                             F                                   T

F                             T                                   T

T                             F                                    F

p if and only if q: Let p and q be two propositions, we define a proposition “p if and only if q” denoted by pq which true if both p and q are true or false and is false if p is true while q is false and p is false while q is true.

p                            q                                 p↔q

F                             F                                   T

F                             T                                   F

T                             F                                   F

T                            T                                    T

Exercise: Let p denote statement “The food is good”, q denote statement “The service is good”, and r denote statement “The rating is three star”.Then write the following statements in symbolic form.

a) Either food is good or the service is good or both.

b) Either food is good or the service is good but not both.

c) The food is good while the service is poor.

d) It is not true case that both the food is good and the rating is three star.

f) It is not true that a three star rating always means that good food good service.

Exercise: Let p be “It is cold” and q be “It is raining”. What do you mean by the following symbols?

Answer: (i) It is cold and raining.

(ii) It is cold or it is raining.

(iii) It is raining if and only if it is cold.

(iv) It is cold and it is not raining then it is cold.

Exercise: Let p denote the statement “The material is interesting”, q denote the statement “The exercise are challenging” and r denote the statement “ The course is enjoyable”. Write the following statement in symbolic form.

1. The material is interesting and the exercise is challenging.