Mathematical Statement: A mathematically acceptable statement is a sentence which is either true or false but not both. For example,
- Two plus two equals four.
- The sum of two positive numbers is positive.
- All prime numbers are odd numbers.
Of these sentences, the first two are true and the third one is false.
Compound Statement: If two or more simple statements are combined using words such as ‘and’, ‘or’, ‘not’, ‘if’, ‘then’, and ‘if and only if’, then the resulting statement is called a compound statement.
A statement can either be ‘true’ or ‘false’ which are called truth values of a statement and these are represented by the symbols T and F, respectively.
A truth table is a summary of truth values of the resulting statements for all possible assignment of values to the variables appearing in a compound statement.
Truth table for two statements (a, b)
The phrases or words which connect simple statements are called logical connectives/ operations or sentential connectives or simply connectives.
|
a |
b |
| T |
T |
| T |
F |
|
F |
T |
| F |
F |
AND Operation(multiplication): A compound sentence formed by two simple sentences p and q using connective ‘and’ is called the conjunction of p and q. It is represented by y = p^q.
| P | q |
p.q |
|
T |
T | T |
| T | F |
F |
|
F |
T | F |
| F | F |
F |
OR Operation (addition operation): A compound statement formed by two simple sentences p and q using connectives ‘or’ is called disjunction of p and q. represented by y = p˅q
| P | q |
p + q |
|
T |
T | T |
| T | F |
T |
|
F |
T | T |
| F | F |
F |
NOT Operation: A statement which is formed by changing the truth value of a given statement by using the word like “no”, ‘not’ is called negation of given statement. If p is a statement, then negation of p is denoted by “∼p”.
|
P |
∼p |
|
T |
F |
| F |
T |
Conditional Operation: Two simple statements p and q connected by the phrase ‘if and then’ is called conditional statement of p and q.
|
P |
q |
p.q |
|
T |
T | T |
| T | F |
F |
|
F |
T | T |
| F | F |
T |
Biconditional Operation: The two simple statements connected by the phrase ‘if and only if’, this is called biconditional statement.
|
P |
q | p.q |
| T | T |
T |
|
T |
F |
F |
|
F |
T | F |
| F | F |
T |
Students get confused when these four terms are played in their minds: Reverse, Converse, Inverse and Contrapositive.
Consider the implication formula.
- Its reverse
- Its converse
- Its inverse
- Its contrapositive
The compound statement which is true for every value of its components is called tautology. For an example, (p q) (q p) is a tautology.
The compound statement which is false for every value of its components is called contradiction/fallacy. For an example, {(p q) (q p)} is a fallacy
Contrapositive: The contrapositive of a statement p ⇒ q is the statement ∼q ⇒ ∼p
Converse: The converse of a statement p ⇒ q is the statement q ⇒ p