**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*) = 2*x *+ 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*) *∈ **A*2 : *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 Cheatsheet**1

1 Resources:http://tutorial.math.lamar.edu/pdf/Algebra_Cheat_Sheet.pdf

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