# MAT 230 MODULE 5

**Module Five Homework**

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**General:**

- Before beginning this homework, be sure to read the textbook readings and the module notes.
- For additional practice, each homework problem has some ungraded examples and sample problems from the text that you can review, and they directly correspond with your graded homework. Work on those problems if needed, and post your questions to this week’s ungraded discussion forum.
- Work within this document for your homework, and be sure to show all steps for arriving at your solution.

**Section 4.1 Homework**

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1) For A = {a, u, p, l, m} and B = {21, 32, 5}

- How many elements are in A´B?
- List the elements of A´

This problem is similar to example 4 and problems 4.1.5–4.1.7.

**Section 4.2 Homework**

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1) Let A = ℤ^{+}, the positive integers, and let *R* be the relation defined by *a R b *if and only if 4*a* > 2b + 3.

- Give two ordered pairs that belong to R.
- Give two ordered pairs that do not belong to R.

This problem is similar to examples 3 and 4 and problems 4.2.1–4.2.3.

( 2,1) ,(5,2) E R . As 4*2 > 2*1+3 etc

(1,2) ,(2,5) does not belong to R you can check easily

2) Let A = {1, 2, 3, 4, 5, 6} = B. Define *a R b *if and only if *a* + *b* > 7. Find the domain, range, matrix representation of R, and the digraph of R.

Examples 22 and 23 deal with the digraph of a relation. This problem is similar to problems 4.2.7 and 4.2.9–4.2.11. Below are images you can use to create the diagraph. You may copy/paste the arrow and alter the directions and length as needed.

**Section 4.4 Homework**

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1) Determine whether R is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. If the relation fails to have a property, give an example showing why it fails in this case.

- Let A = {1, 2, 3, 4} and R be a relation on the set A defined by R = {(1,2)}.
- Let A = {1, 2, 3, 4} and R be a relation on the set A defined by:

R = {(1,1), (2,2), (3,3), (1,2), (2,1), (1,4), (3,2), (3,1), (3,4), (4,2), (4,1)}.

This problem is similar to examples 1c, 4, and 10 and problems 4.4.1–4.4.4, 4.4.7, and 4.4.8.

2) Let A = {1, 2, 3, 4, 5} and R be the relation on the set A whose digraph is:

Determine whether R is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. If the relation fails to have a property, give an example showing why it fails in this case.

This problem is similar to problems 4.4.9 and 4.4.10.

3) Let A = {1, 2, 3, 4, 5} and R be the relation on the set A whose matrix is:

**M _{R}** =

Determine whether R is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. If the relation fails to have a property, give an example showing why it fails in this case.

This problem is similar to example 6 and problems 4.4.11 and 4.4.12.

4) A = ℤ; *a R b *if and only if *a *+ *b *is odd. Determine whether R is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. If the relation fails to have a property, give an example showing why it fails in this case.

This problem is similar to examples 2, 5, 8, and 9 and problems 4.4.13–4.4.19.