Understanding Properties of Functions and Relations in Set Theory

Question 1

Prove that  is irrational?

Question 2

A certain relation * is defined on z² by

(x₁, x₂) *  (y₁, y₂) if and only if there exists a real number k 1 such that y₂ = kx₁ and 

x₂= ky₁.

For each of the following questions, be sure to provide a proof supporting your answer.

Is *  reflexive?

Is *  symmetric?

Is *  anti-symmetric?

Is *  transitive?

Is * an equivalence relation, a partial order, both, or neither?

Question 3

Consider the two functions f :A  Band h : B  C for non-empty sets A, B, C.

Decide whether each of the following statements is true or false, and prove each claim.

If h o f  is surjective, then / is surjective.

If h o f is injective, then / is injective.

If h o f is injective and f is surjective, then h is injective.

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