Existence of Function from various Domains to various ranges.

Question

Multiple correct options:

Which of the following is/are true?

(a) There is a surjective function $f:\mathbb{N} \to \mathbb{N} \times \mathbb{N}.$

(b) There is a injective function $f:\mathbb{N} \times \mathbb{N} \times \mathbb{N} \to \mathbb{N}.$

(c) There is a bijective function $f:\mathbb{N} \times \mathbb{N} \to \mathbb{N} \times \mathbb{N} \times \mathbb{N}.$

(d) There is a bijective function $f:\left\{ 0,1 \right\} \times\mathbb{N} \to \mathbb{N} \times \mathbb{N}.$

I think (b) is only correct since bijective requires the same cardinality.

How should I do this one.

They're all true. All the sets here have the same cardinality. — jgon, Dec 29 '17 at 05:02
Also as a clue. If you can show there is a bijective function $ f : \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N} $ you can use that to prove all the others. — Q the Platypus, Dec 29 '17 at 06:40

score 0 · Answer 1 · answered Dec 29 '17 at 06:51

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All the given alternatives are correct because if there is a bijective function between two sets then that function is surjective as well as injective.

answered Dec 29 '17 at 06:51

D.P.Choudhury

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It would be helpful to the person asking to show that the bijection exists. – Q the Platypus Dec 29 '17 at 09:07

Existence of Function from various Domains to various ranges.

1 Answers1