Mathematics

Problem

Which of the following gives the correct condition for a function

f:A\to B

to be surjective (onto)? a) For every

x\in A

, there exists

y\in B

such that

f(x)=y

. b) For every

y\in B

, there exists a unique

x\in A

such that

f(x)=y

. c) For every

y\in B

, there exists

x\in A

such that

f(x)=y

. d) For all

x_1,x_2\in A

f(x_1)=f(x_2)

implies

x_1=x_2

Intuitive explanation

A function is surjective when every element in the codomain $B$ actually gets "hit" by at least one input from $A$ . The key insight is that surjectivity is about coverage: you're checking whether the range equals the codomain. Instead of thinking about individual mappings, ask yourself whether there's any element in $B$ left out—if no element is left out, the function is surjective.

Intuitive explanation

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