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Question

Show that if f1 and f2 are one-one maps from R to R, then the product f1×f2 : RR defined by f1×f2 x=f1 x f2 x need not be one-one.

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Solution

We know that f1: R → R, given by f1(x) = x, and f2(x) = x are one-one.
Proving f1 is one-one:
Let x and y be two elements in the domain R, such that
f1(x) = f1(y)
x = yet f1(x)=f1(y)x=y
So, f1 is one-one.

Proving f2 is one-one:
Let x and y be two elements in the domain R, such that
f2(x) = f2(y)
x = yet f1(x)=f1(y)x=y
So, f2 is one-one.

Proving f1 × f2 is not one-one:
Given:
f1 × f2x=f1 x × f2 x=x × x=x2Let x and y be two elements in the domain R, such thatf1 × f2x=f1 × f2yx2 = y2x=± ySo, f1 × f2 is not one-one.f1×f2

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