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Injective, surjective, and bijective compositions.

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Let f: X--->Y and g: Y--->Z be functions.

i. If both f and g are injective prove that g o f is injective.
ii. If both f and g are surjective prove that g o f is surjective
iii. If both f and g are bijective prove that g o f is bijective
iv. If g o f is bijection, prove that f is an injection and g is a surjection.

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Solution Summary

This provides examples of working with proofs that compositions of functions are injectie, surfective, and bijective.

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Proof:
Let f: X--->Y and g: Y--->Z be functions.

(1) If both f and g are injection, then f(x) = f(y) implies x = y and g(x) = g(y) implies x = y.
Now we consider any x, y in X and suppose g o f (x) = g o f (y), then g(f(x)) = g(f(y))
Since g is injective, then f(x) = f(y). Since f is injective, then x = y.
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