QUESTION 1 LetX ? Y be a function that maps elements of a set X (domain) to thos

Question

QUESTION 1 LetX ? Y be a function that maps elements of a set X (domain) to those of a set Y (target). which of the following statements is possibly false? a. For every x eX there is a unique y E Y such that f(x)-y. The setf(xy) | xeX and fix) -y ls a subset of X x Y. c. For every y E Y, there is a unique x ? X such that f(x)-y. d.The ronge of f must be a subset of Y QUESTION 2 Consider the functionf: R?R, where R represents the set of all real numbers and for every x (x)-x . which of the following statements is true? a. f is neither one-to-one nor onto. fis onto but not one-to-one. fis one-to-one but not onto. ??. OQ.fis one-to-one and onto. QUESTION 3 Consider the functiong: Z-+ s0, 1, 2, 3, 4, where Z represents the set of al integers and for every x E Z gx)-x mod 5, i.e. gx) is the remainder of x divided by 5. Which of the following statements is true? a, g is neither one-to-one or onto. b.g is one-to-one and onto. o c, g is onto but not one-to-one. d.g is one-to-one but not onto.

Explanation / Answer

1) c.This is true only for certain functions which are onto also called surjective . For functions other than onto functions this statement will be false

2) d. As the domain and range are real numbers, the function will have have one-one relation i.e for every x there will be a unique x^3 associated with it . Also for every element in the range there exits an element in the domain hence its onto also Therefore f is both one-one and onto

3) c. g is not one-one because for different values of input we may get same output for example 5,10 both will have 0 as output and g is onto because for every value of range{0,1,2,3,4} there exists an element in the domain

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