MATH5301 Elementary Analysis. Homework 2.
2.1
For a function f : A → B Show that for any X ⊂ A, Y, Z ⊂ B
(a) X ⊂ f
−1
(f(X))
(b) f(f
−1
(Y )) ⊂ Y
(c) f
−1
(Y ∪ Z) = f
−1
(Y ) ∪ f
−1
(Z)
(d) f
−1
(Y ∩ Z) = f
−1
(Y ) ∩ f
−1
(Z)
2.2
Show that
(a) A ∩
S
λ∈Λ
Aλ =
S
λ∈Λ
(Aλ ∩ A)
(b)
T
λ∈Λ
Aλ
∪
T
λ∈Λ
Bλ
=
T
λ∈Λ
(Aλ ∪ Bλ)
2.3
Which of those are equivalence relations?
(a) for a, b ∈ R, let aRb if a − b ∈ Q.
(b) for a, b ∈ R, let aRb if a − b /∈ Q.
(c) for a, b ∈ R, let aRb if a − b is a square root of rational number.
(d) Let X = Z × N, let x = (x1, x2) and y = (y1, y2) are in R if x1y2 = x2y1.
2.4
For the relation (x, y) (a, b) if (x ≥ a) ∧ (y ≥ b) on the set of ordered pairs of {1, 2, 3} × {1, 2, 3}
(a) Show that the above relation is an order relation.
(b) Can you make it the total order?
(c) How many different total ordering can be constructed?
2.5
Provide an example of f : Z → N such that
(a) f is surjective, but not injective,
(b) f is injective, but not surjective,
(c) f is surjective, and injective,
(d) f is niether surjective nor injective.
2.6 Is the following statement correct?
Theorem 1. If the relation R on A is symmetric and transitive, then it is reflexive.
Proof. For any a ∈ A let b ∈ A is such that aRb. Then, by symmetry bRa. Then by symmetry aRa.
