Question

Consider the set A ={1,2,3}. Write the smallest and largest equivalence relation on A.

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Answer 1

Answer:

set A ={1,2,3}

smallest equivalence relation on set A = { (1,1) (2,2) ( 3,3)}

largest equivalence relation on set A =  { (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) }

Step-by-step explanation:

Equivalence Relation:

an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:

A relation which is reflexive, symmetric and transitive is called "equivalence relation".

iff (a,a) ∈ R ∀ a ∈ A (reflexive property),

iff (a,b) ∈ R ⇒ (b,a) ∈ R ∀ a,b ∈ R (symmetric property), and

iff (a,b) & (b,c) ⇒ (a,c) ∈ R ∀ a, b, c ∈ R then  (transitive property).

Consider an example set, S = (1,2,3)

Equivalence property follows, reflexive, symmetric and transitive

Largest ordered set are s x s = { (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) } which are 9 which equal to 3^2 = n^2

Smallest ordered set are { (1,1) (2,2) ( 3,3)} which are 3 and equals to n. number of elements.