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

In summary, an injection is a function that maps each element of its domain to a unique element in its codomain. The composition of two injections is also an injection, and to prove this, it must be shown that for any two injections f and g, the composition f(g(x)) is also an injection. The mathematical notation for an injection is f: A → B, where A is the domain and B is the codomain. It is important to prove that the composition of injections is an injection because it ensures that the resulting function is one-to-one and has various applications in mathematics and computer science.

Aug 5, 2010

jy02354441

how to prove it please?

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Aug 6, 2010

HallsofIvy

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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, ...

1. How do you define an injection?

An injection is a function that maps each element of its domain to a unique element in its codomain. In other words, each input has only one corresponding output.

2. What is the composition of injections?

The composition of two injections is the function resulting from applying one injection to the output of the other. This means that the composition of injections is also an injection.

3. How do you prove that the composition of injections is an injection?

To prove that the composition of injections is an injection, we need to show that for any two injections f and g, the composition f(g(x)) is also an injection. This can be done by showing that for any inputs x and y, if f(g(x)) = f(g(y)), then x = y.

4. What is the mathematical notation for an injection?

The notation for an injection is f: A → B, where A is the domain and B is the codomain. This can also be written as f(x) = y, where x and y are elements of A and B, respectively.

5. Why is it important to prove that the composition of injections is an injection?

Proving that the composition of injections is an injection is important because it ensures that the resulting function is one-to-one, meaning that each input has a unique output. This property is useful in various areas of mathematics and computer science, such as cryptography and data compression.