Applied Mathematics - Mathematical Reasoning

Description

Mathematical Reasoning

• Mathematically acceptable statements

• Connecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics

• Validating the statements involving the connecting words difference between contradiction, converse and contrapositive

SUMMARY

1. A mathematically acceptable statement is a sentence which is either true or false.

2. Explained the terms:

– Negation of a statement p: If p denote a statement, then the negation of p is denoted by ∼p.

– Compound statements and their related component statements: A statement is a compound statement if it is made up of two or more smaller statements. The smaller statements are called component statements of the compound statement.

– The role of “And”, “Or”, “There exists” and “For every” in compound statements.

– The meaning of implications “If ”, “only if ”, “ if and only if ”. A sentence with if p, then q can be written in the following ways.

– p implies q (denoted by p ⇒ q)

– p is a sufficient condition for q

– q is a necessary condition for p

– p only if q – ∼q implies ∼p

– The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p . The converse of a statement p ⇒ q is the statement q ⇒ p. p ⇒ q together with its converse, gives p if and only if q.

3. The following methods are used to check the validity of statements: (i) direct method (ii) contrapositive method (iii) method of contradiction (iv) using a counter example.

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