Notes/UNB/Year 4/Semester 2/CS2333/2024-01-22.md

Lecture Topic: Relation examples

Relation Example 1

$R = {(i,j) | i \in \mathbb{Z}, j \in \mathbb{Z}, i-j =5n \text{ for some integer n}}$ e.g (22,7) \in R because $22-7 = 5(3)$ e.g (7,22) \in R because $7-22 = -15 = 5(-3)$ e.g (22,9) \notin R because 22-9 = 13, which is not a multiple of 5

Reflexive? Yes For every integer a, $a-a = 0 = 5(0)$ Therefor (a,a) \in R for every integer a

Symmetric? Yes Let a, b be any integer, such that (a,b) is in the relation (can we prove that (b,a) is in relation as well?)

Since (a,b) is in the relation, a-b = 5n, so, $b-a = -5(n) = 5(-n)$ Note: (b-a) = -(a-b)

Transitive? Yes Let a,b,c be any integers such that (a,b) \in R and $(b,c) \in R$ Can we prove that (a,c) \in R ?

Since (a,b) \in R, we know a-b=5n for some $n \in \mathbb{Z}$ Since (b,c) \in R, we know b-c=5p for some n \in \mathbb{Z}

Now, $(a-c) = (a-b) + (b-c) = 5n + 5p = 5(np)$ Therefore, $(a-c) \in R$ Note: This is an integer because it is the sum of integers

Relation Example 2

R = \{(i,j) | i \in \mathbb{R}, j \in \mathbb{R}, i + 2 > j\}

Examples:

• (5,2) \in R because (5+2) > 2
• (3,4) \in R because (3+2) > 4
• (0,7) \notin R because (0+2) \ngtr 7

Reflexive? Yes Symmetric? No Transitive? No

Proof: We need to find real numbers where a,b,c where (a,b) \in R and (b,c) \in R but (a,c) \notin R

For example, a = 1, b = 2, c = 3 (1,2) \in R because $1+2 > 2$ (2,3) \in R because 2+2 > 3

But (1,3) \notin R because $(1+2) \ngtr 3$ \therefore R is not transitive

Equivalence relations

Equivalence relations have all three of these properties Any equivalence relation R on a set A induces a partition of A

• Splits A into equivalence classes
  • Each class contains elements that are related to themselves and to each other, but to nothing outside the class

So, for the above relation:

R = \{(i,j) | i \in \mathbb{Z}, j \in \mathbb{Z}, i-j =5n \text{ for some integer n}\}

This forms an equivalence relation

So for the set of all integers, this forms 5 equivalence classes:

• ..., 0, 5, 10, ...
• ..., 1, 6, 11, ...
• ..., 2, 7, 12, ...
• ..., 3, 8, 13, ...
• ..., 4, 9, 14, ...

Example

$A = {-3, -2, -1, 0, 1, 2, 3}$ R = \{(i,j) | i \in \mathbb{A}, j \in \mathbb{A}, i^2 = j^2\}

So the equivalence classes induced by this relation:

• -3, 3
• -2, 2
• -1, 1
• 0