Understanding Properties of Functions and Relations in Set Theory

Question1

Prove that isirrational?

Question2

Acertainrelation*isdefinedon z² by

(x₁,x₂)*(y₁,y₂)ifandonlyifthere existsareal numberk 1 suchthaty₂=kx₁and

x₂=ky₁.

For eachofthefollowingquestions,besuretoprovideaproofsupportingyouranswer.

Is*reflexive?

Is*symmetric?

Is*anti-symmetric?

Is*transitive?

Is*anequivalencerelation,apartialorder,both,orneither?

Question3

Considerthetwofunctionsf:ABandh:BCfornon-emptysetsA,B,C.

Decidewhethereachofthefollowingstatements istrueorfalse,andproveeachclaim.

Ifhof issurjective, then /issurjective.

Ifhofisinjective,then/isinjective.

Ifhofisinjectiveandfissurjective,thenhisinjective.

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