CS 2100: Discrete Structures
Practice Quiz 4
Fall 2020
离散结构quiz代写 Prove R is an equivalence relation on A, or give a counterexample. If R is an equivalence relation, what is the partition of A induced by R?
Warning: This quiz is as long as the actual Quiz 4. However, it does not cover all of the materials of the actual Quiz 4.
1 .(5 minutes) Prove that the function f : Z → Z with f(x) = 2x + 3 is one-to-one.
2 .(5 minutes) Prove for any relation R on a set A, if R = R−1 , then R is symmetric.
3. (8 minutes) Prove or provide a counterexample to the proposition be-low:
The relation R on Z given by R = {(a, b) : a = b} is symmetric and transitive.
4. (10 minutes)
Prove or provide a counterexample to the propositions below: 离散结构quiz代写
The relation R on S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} given by R ={(a, b) ∈ S × S : a|b} is
(a) Reflexive
(b) Antisymmetric
(c) Transitive
(d) Is R a strict order on S (Yes/No)?
(e) Is R a partial order on S or a total order? If R is a partial order,draw its Hasse Diagram. If R is a total order, which element(s)would need to be removed so that R would be a partial order?
For each question, provide detailed proofs/justifications.
5.(15 minutes) Let A = {0, 1, 2, 3, 4, 5} and B = {0, 1, 2, 3}
(a) There is no possible function f : A → B that is one-to-one. (True/False)
(b) There is no possible function f : A → B that is onto. (True/False)
(c) g = {(0, 0),(1, 2),(2, 1),(4, 3),(5, 3)} is a function from A to B.(True/False)
(d) Let R be a relation with domain B and codomain B, and R ={(0, 1),(1, 2),(2, 3),(3, 3),(3, 0)}:
i.Is R Transitive? (Yes/No)
ii.Is R Antisymmetric? (Yes/No)
iii. Is R Reflexive? (Yes/No)
iv.Is R ◦ R antisymmetric? (Yes/No)
(e) Let R be a relation with domain A and codomain A where (x, y) ∈ R if x mod 3 = y. Is R Antisymmetric? (Yes/No)
6.(6 minutes) Let f : R → R, where f(x) = x 2 .
(a) Prove f is one-to-one, or provide a counterexample:
(b) Prove f is onto, or provide a counterexample:
(c) Is f a one-to-one correspondence? (Yes/No)
For each question, provide detailed proofs/justifications.
7.(10 minutes) Consider the relation R on P({x, y, z}) given by R ={(A, B) : A ∪ B = {x, y, z}}. 离散结构quiz代写
[(a)]
(a) Prove R is reflexive, or provide a counterexample.
(b) Prove R is symmetric, or provide a counterexample.
(c) Prove R is transitive, or provide a counterexample.
(d) Is R an equivalence relation? (Yes/No)
For each question, provide detailed proofs/justifications.
8.(10 minutes) Let A = {1, 2, 3, 4}. Functions f and g are defined as below.
f : A → (A × A) with the rule f(a) = (a, a)
g : (A × A) → A with the rule g(x, y) = x
Are f and g inverses of each other? (Yes/No)
Provide a detailed proof of your decision.
Hint: checking f ◦ g and g ◦ f.
9. (10 minutes) Let A = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, and let a relation R on A be defined with the rule: 离散结构quiz代写
R = {(a, b) ∈ A2 : a and b have the same set of prime factors}.
For example, 2 and 8 have the same set of prime factors: {2}.
2 and 12 do not have the same set of prime factors; the prime factors of 12 are {2, 3}.
Prove R is an equivalence relation on A, or give a counterexample. If R is an equivalence relation, what is the partition of A induced by R?
Basic Algebra Cheatsheet1
1 Resources:http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet.pdf
更多代写：数学课程辅导 gre远程代考 商科网课作业 scholarship essay写法 心理学论文英文 化学作业代写
合作平台：essay代写 论文代写 写手招聘 英国留学生代写
