# Mathematical Reasoning

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.

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