Virtual Labs

Equivalence Relation

Relation

R

is defined as

R = \{(a, b) \mid (a - b) = k \cdot m \text{ for some fixed integer } m \text{ and } a, b, k \in \mathbb{Z}\}

, then

R

a: Reflective but not symmetric Explanation

Explanation

b: Symmetric but not transitive Explanation

Explanation

c: Transitive but not reflective Explanation

Explanation

d: An equivalence relation Explanation

Explanation

A relation

R

is said to be circular if

aRb

and

bRc

together imply

cRa

. Which of the following options is/are correct?

a: If a relation

S

is transitive and circular, then

S

is an equivalence relation. Explanation

Explanation

b: If a relation

S

is reflexive and symmetric, then

S

is an equivalence relation. Explanation

Explanation

c: If a relation

S

is reflexive and circular, then

S

is an equivalence relation. Explanation

Explanation

d: If a relation

S

is circular and symmetric, then

S

is an equivalence relation. Explanation

Explanation

Suppose a relation

R = \{(3, 3), (5, 5), (5, 3), (3, 5), (6, 6)\}

S = \{3, 5, 6\}

. Here

R

is known as

a: equivalence relation Explanation

Explanation

b: reflexive relation Explanation

Explanation

c: symmetric relation Explanation

Explanation

d: transitive relation Explanation

Explanation

(a, b) \in R \implies (b, a) \in R

\forall a, b \in A

then

R

a: Symmetric Explanation

Explanation

b: Transitive Explanation

Explanation

c: Reflexive Explanation

Explanation

d: Equivalence relation Explanation

Explanation

(a, b) \in R

and

(b, c) \in R \implies (a, c) \in R

\forall a, b, c \in A

then

R

a: Symmetric Explanation

Explanation

b: Transitive Explanation

Explanation

c: Reflexive Explanation

Explanation

d: Equivalence relation Explanation

Explanation

Let

R

be a relation on set

A

where

|A| = n

. What is the minimum number of ordered pairs needed for

R

to be an equivalence relation?

a: n Explanation

Explanation

b: n + 1 Explanation

Explanation

c: n² Explanation

Explanation

d: 2n Explanation

Explanation

Given a relation

R

on set

A

, if

R

is symmetric and transitive, what additional property must

R

have to be an equivalence relation?

a: Antisymmetric Explanation

Explanation

b: Reflexive Explanation

Explanation

c: Irreflexive Explanation

Explanation

d: Asymmetric Explanation

Explanation

R

is an equivalence relation on set

A

, and

[a]

denotes the equivalence class of

a

, which statement is always true?

[a] \cap [b] = \emptyset

a \neq b

Explanation

[a] = [b]

if and only if

(a,b) \in R

Explanation

[a] \cup [b]

is always an equivalence class Explanation

Explanation

|[a]| = |[b]|

for all

a,b \in A

Explanation

Let

R

be an equivalence relation on an infinite set

A

. Which statement must be true about the cardinality of equivalence classes?

a: All equivalence classes must be finite Explanation

Explanation

b: All equivalence classes must be infinite Explanation

Explanation

c: At least one equivalence class must be infinite Explanation

Explanation

d: The equivalence classes can be all finite or all infinite or mixed Explanation

Explanation

R

and

S

are equivalence relations on a set

A

, what can be said about their intersection

R \cap S

a: It must be an equivalence relation Explanation

Explanation

b: It is always reflexive but may not be transitive Explanation

Explanation

c: It is always symmetric but may not be reflexive Explanation

Explanation

d: It may not be an equivalence relation Explanation

Explanation

Let

R

be an equivalence relation on a finite set

A

with

n

elements. If there are exactly

k

distinct equivalence classes, what can be said about their sizes?

a: All equivalence classes must have size n/k Explanation

Explanation

b: The sum of the sizes must equal n Explanation

Explanation

c: Each equivalence class must have at least k elements Explanation

Explanation

d: The product of the sizes must equal n Explanation

Explanation