Multiple exercises. 1) Circle T or F, depending on the truth value of the follow
ID: 3108597 • Letter: M
Question
Multiple exercises.
1) Circle T or F, depending on the truth value of the following statements. For statements with quanties the universal set is R.
a) The relation on R given by x R y if xy > 0 is an equivalence relation.
T F
b) If the set A is nite and admits an equivalence relation,
then all equivalence classes have the same number of elements.
T F
c) The function f : N N dened by f(x) = x + 1 is surjective.
T F
d) The intersection of two equivalence relations is an equivalence relation.
T F
e)
Every bijection is surjective.
T F
f)
The empty relation is symmetric.
T F
g) The function f : N N dened by f(x) = x + 1 is injective.
T F
h)
All sets admit at least one partition.
T F
i)
The relation is an equivalence relation on Z.
T F
j) Under the relation a b (mod 4), the set Z has 4 distinct equivalence classes.
T F
2) a) Let S = {1, 2, 3, 4}, and dene a relation
R := {(1, 2), (1, 3), (2, 3), (1, 4), (4, 1)}.
Write down a relation R1 such that R R1, R1 is transitive, and no proper subset of R1 that contains R is transitive.
b) Write down all possible equivalence relations on the set {a, b}.
3)
Consider the following sequence a1, a2, a3, ... dened by
a1 = 1, a2 = 2, and an = 2an 1 an 2 for n 3.
Use induction to prove that an = n for all n N.
4) Suppose A, B, and C, are sets, and that f : A B and g : B C are functions. If g f is surjective, is f surjective? Prove or give a counter example.
5) We dene a relation on the set Z by
x y if x2 y2 = 4k
for some k Z. Prove is an equivalence relation, and describe its equivalence classes.
Explanation / Answer
ANSWER :-
a) The relation on R given by x R y if xy > 0 is an equivalence relation.
Answer :- FALSE
b) If the set A is nite and admits an equivalence relation,then all equivalence classes have the same number of elements.
Answer :- FALSE
c) The function f : N N dened by f(x) = x + 1 is surjective.
Answer :- FALSE
d) The intersection of two equivalence relations is an equivalence relation.
Answer :- TRUE
e) Every bijection is surjective.
Answer :- TRUE
f) The empty relation is symmetric.
Answer :- TRUE
g) The function f : N N dened by f(x) = x + 1 is injective.
Answer :- TRUE
h) All sets admit at least one partition.
Answer :- TRUE
i) The relation is an equivalence relation on Z.
Answer :- FALSE
j) Under the relation a b (mod 4), the set Z has 4 distinct equivalence classes.
Answer :- TRUE
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