Mathematical Reasoning

A mathematical sentence is a sentence that states a fact or contains a complete idea. A sentence that can be judged to be true or false is called a statement or a closed sentence.
The negation of a statement is a statement with the opposite meaning. It is usually constructed by adding a 'not' or removing a 'not' from the statement.
Some of the connecting words which are found in compound statements like 'And', 'Or’ are often used in mathematical statements. These are called connectives.
Quantifiers are the words that are used to indicate the number of objects or cases in a statement. 'There exists' and 'For every' are the two phrases known as quantifiers.
The statements of the form 'if p then q', 'p only if q' and 'if and only if' are called implications.
Given an if-then statement 'if p, then q', we can create other related statements: To form the converse of the conditional statement, interchange the antecedent and the consequent.
Here, we have learned, how to check the validity of a statement, i.e., how to check when it is true and when it is not true.
The truth and falsity of the statement depends upon the following:
Special words: 'and', 'or'
Implications: 'if-then', 'if and only if'
Quantifiers: 'for every', 'there exists'

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  • Q1

    Write the converse of the following statement:
    If x is a prime number then x is odd.


    If a number x is odd then x is prime.

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  • Q2

    Define quantifiers.


    Quantifier are phrases like "There exists" and "For every ( or for all)".
    Depending upon the context the phrase "There exists" can also be replaced by the equivalent phrase "There is at least one"

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  • Q3

    Rewrite the following statement in the form of “if p, then q” : A quadrilateral is a rectangle if it is equiangular.


    Here p is: A quadrilateral is equiangular

    and q is: A quadrilateral is a rectangle.

    Rewriting it in the form “if p then q” we get

    If a quadrilateral is equiangular, then it is a rectangle.

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  • Q4

    Rewrite the following statement in the form of “p if and only if q”.
    If you watch television then your mind is free and if your mind is free then you watch television.


    Here p : you watch television
    and q : your mind is free.
    Rewriting it in the form “ p if and only if q” we get
    You watch television if and only if your mind is free.

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  • Q5

    Write the negation of the following statement “The number 8 is greater than 7.”


    The negation of the statement is:

    The number 8 is not greater than 7.

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