Math 8 Equivalence Relation and Set Theory Worksheet Number 8 please do problem 3(b) and 6 7 8. Please proving the problem by using formal mathmatics logic and symbol thx. Questions are on the pdf above need finish in 2hours!!!!!!!!!!!!! Math 8

Homework 6

1. Write vectors in R2 as (x, y). Define the relation on R2 by writing (x1 , y1 ) ∼ (x2 , y2 ) iff

y1 − sin x1 = y2 − sin x2 . Prove that ∼ is an equivalence relation. Find the classes [(0, 0)],

[(2, π/2)] and draw them on the plane. Describe the sets which are the equivalence classes for

this relation.

2. Let ∼1 and ∼2 be two equivalence relations on S. Define for a, b ∈ S

n

o

a ∼ b ⇔ a ∼1 b and a ∼2 b .

Prove that ∼ is an equivalence relation on S. For a fixed x ∈ S find the formula for [x]∼ in

terms of [x]∼1 and [x]∼2 . Prove your formula.

3. Let S be a set and x0 ∈ S. Let P(S) be the power set of S. Prove that the following are not

equivalence relations on P(S):

(a) for A, B ⊆ S define A ∼ B if and only if A ∩ B 6= ∅;

(b) for A, B ⊆ S define A ∼ B if and only if x0 ∈ A ∩ B.

4. Let f : A → B be a map. A level set is the set f −1 ({z}) for some z ∈ B. Spell out the

definition of f −1 ({z}) and explain in your words why it is called level set.

Define for a, b ∈ A

a ∼ b ⇔ f (a) = f (b).

Prove that ∼ is an equivalence relation on A. Describe A/ ∼ and the equivalence classes [a]

using the level sets.

5. List all functions f : A → B with A = {0, 1, 2} and B = {a, b} (there are 8 of them).

6. Find the (exact) range Ran(f ) and the (maximal) domain Dom(f ) for the following functions

f : Dom(f ) → Ran(f )

√

(a) x 7→ x + x2 − 1;

(b) x 7→ 1/(x2 + 1);

2

(c) x 7→ e−x .

7. Give an example of maps f and g such that f ◦ g is defined while g ◦ f is not.

In the previous problem denote functions by f1 in (a), f2 in (b), and f3 in (c). Which mappings

below are defined? Write formulas for those mappings which are defined

f1 ◦ f2 , f1 ◦ f3 , f2 ◦ f1 , f2 ◦ f3 , f3 ◦ f2 .

8. For every function f in problem 6 above find: f ([−1, 1]), f ({−1, 1}), f −1 ({4}), f −1 ([−2, −1]).

9. For a map f : A → B and U, V ⊆ B prove that

f −1 (U ∩ V ) = f −1 (U ) ∩ f −1 (V ) and f −1 (U ∪ V ) = f −1 (U ) ∪ f −1 (V ).

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