Suppose S and T are infinite sets and φ: S→T is a function that is not onto. May we conclude that S and T do not have the same cardinality? Prove your answer!

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A: Please post the multiple questions separately. Here I answered question (1) only as per our policy.

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A: option (c) is correct.

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Q: (b) Let A = (x ER 0<x< 1) and B=(x ER | 0<x<2). Determine whether A and B have the same cardinality.

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

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A: The given question is solved below.

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A: As two discrete spaces are homeomorphic iff they have same cardinality .

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Q: (b) Let A = (x ER 0<x< 1} and B = {x ER|0<x<2}. Determine whether A and B have the same cardinality.

A: Solution:-

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A: (a) False f:E → INLet E= IR  Thenf: IR → INdefened as f(x) =  1  , ∨ x ∈ IRi.e There exist a…

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Q: Exercise 4.21. Prove. (1) The set of all functions f : R → R has the same cardinality as : 2R.

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

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Q: 1. If f:X → Y and g:Y → Z are functions and gof is onto. a. Must g be onto? Give a proof or a…

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Q: Suppose f : N → N and g : N → N are functions such that f is not onto but f + g is onto. May we…

A: Given: Two functions f : N → N and g : N → N are functions such that f is not onto but f + g is…