How to Write and 'If and only If' Proof: x^2 is even if and only if x is even

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How to Write and "If and only If" Proof: x^2 is even if and only if x is even. This is a good example for someone learning to write proofs.

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I really like this video because it is quick and a great demonstration of important techniques helpful to beginners.

econhelp

If p then q. The contrapositive is if ~q then ~p.
If x^2 is even then x is even. Or if x is odd then x^2 is odd.
x=2k+1 then x^2= 2(2k^2+2k)+1 yes it is also odd.

lingzhao

I recently started reading a book you recommended “Mathematical proofs; A transition to higher mathematics”, where i learned how to prove something like this. The book is very good and after being stuck in “intermediate math” for a few years now, I feel like this book is a doorway to advanced math.

lukaobradovic

I read a good book that covers this by velleman. His proofwriting book covers all of sentential and quantificational logic in the first 2 chapters before touching proofs!

ronstudios

Your math and reasoning is rather correct, but the use of a 2 column proof where each step has a conditional statement as its justification, makes the argument much clearer. This type of direct proof would be based on the repeated use of the reasoning postulate: " P implies Q and P" implies "Q". This direct proof also makes use of the Transitive Theorem for an Implication.

amtrakatsfnyc

Plz done problems of Larson calculus-
Putnam exam challenge

happy.

proof that every power of a numer with a last digit is 6 ends with a 6, and 5 ends with 25 can be nice too

HanzoHimemiya

A different method of proof is to show that x^2-x is always even.

pedroteran

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