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3.Prove that the Greatest Integer Function f: R → R, givenone-one nor onto, where [x] denotes the greatest integerby fox)-x], is neitherless than or equal to x |
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Answer» A functionf:X→Yf:X→Ywhere for everyx1,x2∈X,f(x1)=f(x2)⇒x1=x2x1,x2∈X,f(x1)=f(x2)⇒x1=x2is called a one-one or injective function. A functionf:X→Yf:X→Yis said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for everyy∈Yy∈Y, there exists an element x in X such thatf(x)=yf(x)=y. Givenf:R→Rf:R→Rdefined byf(x)=[x],f(x)=[x],where[x][x]denotes the greatest integerleqxleqx: Step1: Injective or One-One function For example, if we considerx=1.1x=1.1, we see thatf(1.1)=1f(1.1)=1. Similarly, if we considerx=1.9x=1.9, we see thatf(1.9)=1f(1.9)=1. ⇒[1.1]=[1.9]⇒[1.1]=[1.9], but given1.1≠1.91.1≠1.9, f:R→Rf:R→Rdefined byf(x)=[x],f(x)=[x],is not one-one. |
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