the sentence a given connecting word is introduced. For example, compare the following two sentences: . For every positive number x there exists a positive number y such that y < x. .
There exists a positive number y such that for every positive number x, we have y < x . Although these statements may look similar, they do not say the same thing. As a matter of fact, ( ) is true and ( ) is false. Thus, in order for a piece of mathematical writing to make sense, all of the symbols must be carefully introduced and each symbol must be introduced precisely at the right place – not too early and not too late.
The words “And” and “Or” are called connectives and “There exists” and “For all” are called quantifiers. Thus, we have seen that many mathematical statements contain some special words and it is important to know the meaning attached to them, especially when we have to check the validity of different statements. EXERCISE . .
For each of the following compound statements first identify the connecting words and then break it into component statements. All rational numbers are real and all real numbers are not complex. Square of an integer is positive or negative. (iii) The sand heats up quickly in the Sun and does not cool down fast at night.
(iv) x = and x = are the roots of the equation 3x – x – = . MATHEMATICAL REASONING . Identify the quantifier in the following statements and write the negation of the statements. There exists a number which is equal to its square.
For every real number x , x is less than x + . (iii) There exists a capital for every state in India. . Check whether the following pair of statements are negation of each other.
Give reasons for your answer. x + y = y + x is true for every real numbers x and y. There exists real numbers x and y for which x + y = y + x. .