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Question

Give examples of two surjective function $${f}_{1}$$ and $${f}_{2}$$ from $$Z$$ to $$Z$$, such that $${f}_{1}+{f}_{2}$$ is not surjective.


Solution

A function $$f:A\rightarrow B$$ is said to be a onto function or surjective if every elemnt of $$A$$ i.e, if $$f(A)=B$$ or range of $$f$$ is the co-domain of $$f.$$

So, $$f:A\rightarrow B$$ is surjective for each $$b\in B,$$ there exists $$a\in B$$ such that $$f(a)=b$$
Let $$f_1:Z\rightarrow Z$$ and $$f_2:Z\rightarrow Z$$ be two functions given by
$$\Rightarrow$$  $$f_1(x)=x$$
$$\Rightarrow$$  $$f_2(x)=-x$$
From above function it is clear that both are surjective functions.

Now,
$$\Rightarrow$$  $$f_1+f_2:Z\rightarrow Z$$
$$\Rightarrow$$  $$(f_1+f_2)(x)=f_1(x)+f_2(x)$$
$$\Rightarrow$$  $$(f_1+f_2)(x)=x-x$$
$$\Rightarrow$$  $$(f_1+f_2)(x)=0$$
Therefore, $$f_1+f_2:Z\rightarrow Z$$ is a function given by
$$\Rightarrow$$  $$(f_1+f_2)(x)=0$$
Since, $$f_1+f_2$$ is a constant function, hence it is not an onto or surjective function.

Mathematics

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