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Give examples of two functions `f: N->Z" and "g: Z->Z`such that gof is injective but g is not injective. (Hint: Consider `f(x) = x" and "g(x) = |x|`) |
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Answer» A function `f(x)` is injective when, `f(x_1) = f(x_2)` only if `x_1 = x_2`. Let `f(x) = x and g(x) = |x|`. Here, `g(1) =g(-1) = 1`. So, `g(x)` is not injective. Now, `gof = gof(x) = g(f(x)) = g(x) = |x|` But, `f: N->Z`. It means `x` can not be negative, so, `gof(x_1) = gof(x_2)` only when `x_1 = x_2`. `:. gof` is injective. |
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