CSCI 301 L14 Worksheet

Lecture 14 - Exercises

Part A - Relations

Let \(A = \{1, 2, 3\}\).

1. Write the relation representing \(=\) on the set \(A\); write your answer in set form.

2. Draw a diagram representing the relation \(\le\) on \(A\).

3. Write the relation representing “has the same parity as” on \(A\); write your answer in set form.

4. Consider the relation \(R = \varnothing\) on \(A\). Draw a diagram representing this relation.

Part B - Properties of Relations

1. Determine which of the following relations on the set \(A = \{a, b , c\}\) is reflexive.

   1. \(R=\{(a,a),(b,b), (c,c)\}\)
   2. \(R=\{(a,a), (c,c)\}\)
   3. \(R = \{(a,a),(b,a), (a,c),(c,a), (b,b), (c,c)\}\)
   4. \(R = \varnothing\)

2. Determine which of the following relations on the set \(A = \{a, b , c\}\) is symmetric.

   1. $R={(a,b),(b,a), (a,c),(c,a), (b,c), (c,b)} $
   2. \(R=\{(a,b),(b,a), (a,c),(c,a)\}\)
   3. \(R=\{(a,a), (b,b), (c,c)\}\)
   4. \(R = \varnothing\)

3. Determine which of the following relations on the set \(A = \{a, b, c\}\) is transitive.

   1. \(R=\{(a,b),(b,a), (a,c),(c,a), (a,a), (b,b), (c,c)\}\)
   2. \(R=\{(a,b),(b,a), (a,c),(c,a), (a,a)\}\)
   3. \(R=\{(a,a),(b,b), (c,c)\}\)
   4. \(R= \varnothing\)

4. Fill in the following table regarding relations on \(\mathbb{Z}\):

   

Part C - Equivalence Relations and Equivalence Classes

Let \(A = \{-1, 1, 2, 3, 4\}\).

1. Consider the relation \(R_1 = \{(-1, 1), (1,1), (2, 2), (3, 3), (4, 4)\}\). Notice that this is the “equals” (\(=\)) relation.
   1. Give the members of each of the following equivalence classes:
      • \([-1]\)
      • \([2]\)
      • \([4]\)
   2. How many equivalence classes does \(R_1\) have over \(A\)?
2. Consider the relation \(R_2\), which is “has the same parity as” on the set \(A\).
   1. Draw the relationship in diagram form.
   2. Give the members of each of the following equivalence classes:
      • \([-1]\)
      • \([2]\)
      • \([4]\)
   3. How many equivalence classes does \(R_2\) have over \(A\)?
   4. What are the equivalence classes?
3. Describe (in words is fine) the equivalence classes of the equivalence relation \(\equiv \pmod{3}\).
4. Let \(A=\{a,b,c,d,e\}\). Suppose \(R\) is an equivalence relation on \(A\). Suppose \(R\) has two equivalence classes, and we know that \(aRd\), \(bRc\) and \(eRd\). Draw a complete diagram of \(R\).
5. Define a relation \(R\) on \(\mathbb{Z}\) such that \(xRy\) if and only if \(3x−5y\) is even. Prove \(R\) is an equivalence relation. Describe its equivalence classes.