Week 13 for Students.pdf - Relations Definition. A...

Relations
Definition. A relation , 𝑅 , from a set 𝐴 to a set 𝐵 is a subset 𝑅 ⊆ 𝐴 × 𝐵 .
For now, we will focus on relations on the same set, that is, 𝐵 = 𝐴 .
1

Relations on a set and 3 possible properties
Definition. A relation on a set 𝑨 is a subset 𝑅 ⊆ 𝐴 × 𝐴 . It is common to
abbreviate the statement (𝑎, 𝑏) ∈ 𝑅 as 𝑎𝑅𝑏 .
Suppose R is a relation on a set A .
1. Relation R is reflexive if 𝑥𝑅𝑥 for every 𝑥 ∈ 𝐴 .
i.e., if ∀𝑥 ∈ 𝐴, 𝑥𝑅𝑥 .
(Examples. ≤, ≥, =, |, ≡, ⊆ ; Not reflexive: <, >, ≠, ∤, ≢, ⊈ )
2. Relation R is symmetric if 𝑥𝑅𝑦 implies 𝑦𝑅𝑥 for all 𝑥, 𝑦 ∈ 𝐴 .
i.e., if ∀𝑥, 𝑦 ∈ 𝐴, 𝑥𝑅𝑦 ⇒ 𝑦𝑅𝑥 .
(Examples. =, ≠, ≡, ≢ ; Not symmetric: <, ≤, >, ≥, |, ∤, ⊆ )
3. Relation R is transitive if 𝑥𝑅𝑦 and 𝑦𝑅𝑧 implies 𝑥𝑅𝑧 for all 𝑥, 𝑦, 𝑧 ∈ 𝐴 .
i.e., if ∀𝑥, 𝑦, 𝑧 ∈ 𝐴, ((𝑥𝑅𝑦) ∧ (𝑦𝑅𝑧)) ⇒ 𝑥𝑅𝑧 .
(Examples. <, ≤, >, ≥, |, ∤, ⊆ ; Not transitive: 𝑅 = { 1,2 , 2,3 } .
2

Graphical Ways to Describe Properties
3

Here is a diagram for a relation on a set
Write the sets and . Determine if is reflexive,
symmetric, and/or transitive.
.
4

Let 𝐴 = {1,2,3,4,5,6} . Write out the relation 𝑅 that expresses | (divides)
on 𝐴 . Then illustrate it with a diagram. Determine if 𝑅 is symmetric,
reflexive, and/or transitive.
5

Equivalence Relations
Definition. A relation R on a set 𝐴 is an equivalence relation if it is reflexive,
symmetric, and transitive.
Examples: =, ≡ , “has the same parity (odd/even)”, “has the same sign(+/-)”, “has