Mapping

Question

State true or false

If gof is surjective, then both g and f are surjective.

Is it true or false

See the theorem is

If gof is surjective, then  g is surjective and f need not to be

but need not to be doesnot mean, should not be. rt?

Comments

If gof is surjective, then  g must be surjective.

If gof is injective, then  f  must be injective.

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both g ?

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i copied from question

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means it should be true , right?

but ans given false :(

Somewhere I read it is true

Now confused

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it's false statement, because if you say it is true then f and g have to be surjective every time.

Because we have a counter example against the above statement, hence, it's false. because we have one such instance which proves it incorrect.

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for gof to be surjective only g is needed to be onto and f can be injective or surjective

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• Srestha, if g is surjective and f is surjective then gof and fog surjective.

but

1. if gof is surjective then g must be surjective . cannot say about f g may be or may not be
2. if fog surjective then  f must we surjective ,can not say about g may be or may not be

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See

C is surjective or onto means

every element of C mapped from some element of B

And if B is onto, then every element of B has a mapping from A

This doesnot violates previous condition. So why not it is true?

Need not to be true for every case

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If gof is surjective, then both g and f are surjective.

means if g is surjective and f is surjective then gof  surjective. we know this is true.

now inverse of this is true ? no but contrapossitive is true i.e. If gof is not surjective, then both g or f are not surjective.

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If gof is surjective, then both g and f are surjective.

here and is given so it say both must be onto .