Math for Computer Science - Certification Exam

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I am looking for some much needed help please with one question on the Computer Science Path, Math for Computer Science Exam, Part 2.

The question states: Fill in the blanks to match each statement with the type of proof strategy that should be used to prove, or disprove, that statement.

  1. Prove that all the set of numbers from 1 to 10 are less than 11.
    Proof Strategy: ________
  2. Prove that all numbers less then 5 are odd.
    Proof Strategy: ________
  3. Prove that, if a graph has “n” vertices, it has “n-1” edges.
    Proof Strategy: ________
    The options are: Contrapositive, Counterexample, Induction, Direct, Existence, Exhaustion (Once an option is selected, it cannot be used elsewhere in the response).

I´ve tried this exam 3 times and I haven’t be abble to pass it, I’ve already check the lesson and articles neither way I think this question can have multiple answers, Theres’s another post about this but has been closed and the answer was sent by DM
I really need to complete this exam ASAP due my membership is soon to expire

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Hi. This is strange

Prove that all numbers less then 5 are odd.

They are not. This is trivial but there are multiple strategies to be done here, counter example is enough, but contradiction is fun.
Proof by contradiction:
Assume that all numbers less then 5 are odd.
Clearly, 4 < 5. Which would mean we are assuming 4 is odd. Then that means that for some integer k, 4 = 2k + 1.
Subtract 1 to both sides to get 3 = 2k => 1.5 = k. But k is supposed to be an integer, so we have a contradiction. Therefore 4 is not odd, and therefore not all the numbers less than 5 are odd.

I invite you to read this book and do some of the problems here: https://www.people.vcu.edu/~rhammack/BookOfProof/

It will more than cover basic logical thinking for discrete math for computer science.

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