How to prove that the composition of injection is an injection?

Thread starter jy02354441
Start date Aug 5, 2010
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Composition Injection

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The composition of two injective functions, denoted as F = f(g(x)), is also injective. This is established by proving that if F(x) = F(y), then f(g(x)) = f(g(y)). Since f is injective, it follows that g(x) = g(y). Given that g is also injective, it concludes that x = y, thereby confirming that F is injective. This proof relies on the fundamental property of injective functions where distinct inputs yield distinct outputs.

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Aug 5, 2010

jy02354441

how to prove it please?

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Aug 6, 2010

HallsofIvy

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Homework Helper

F(x) is "injective" if and only if f(x)= f(y) implies x= y.

Suppose both f(x) and g(x) are injective and F= f(g(x))

If F(x)= F(y) then f(g(x))= f(g(y)) so, since f is injective, we have g(x)= g(y). Now, since g is injective, ...