Equivalence relations - One Line Questions
1.
If R is an equivalence relation on a set A, then the equivalence class of an element 'a' is denoted by: —
[a]
2.
Let R be an equivalence relation on a set A. If a R b, then the equivalence class of a, [a], is equal to: —
[b]
3.
Let R be a relation on set A. If R is reflexive, symmetric, and transitive, then R is called: —
An equivalence relation
4.
Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. Which property must be added to R to make it an equivalence relation? —
Add (1, 3) and (3, 1)
5.
Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2)}. Which property must be added to R to make it an equivalence relation? —
Add (2, 1)
6.
Let A be a non-empty set. The empty relation {} on A is: —
Neither reflexive, symmetric, nor transitive
7.
The relation 'is a factor of' on the set of positive integers is: —
Reflexive and transitive but not symmetric
8.
A relation R on a set A is reflexive if: —
For every a ∈ A, (a, a) ∈ R
9.
A relation R on a set A is symmetric if: —
For every a, b ∈ A, if (a, b) ∈ R, then (b, a) ∈ R
10.
A relation R on a set A is transitive if: —
For every a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
11.
Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3)}. Is R an equivalence relation? —
Yes, it is an equivalence relation.
12.
Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. Is R an equivalence relation? —
No, it is not transitive.
13.
Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}. Is R an equivalence relation? —
No, it is not transitive.
14.
Let R be the relation on the set of integers Z defined by a R b if a + b is even. Is R an equivalence relation? —
Yes, it is an equivalence relation.
15.
Let R be the relation on the set of integers Z defined by a R b if a - b is an even number. Is R an equivalence relation? —
Yes, it is an equivalence relation.
16.
Consider the relation R on the set of real numbers R, where x R y if x - y is an integer. Is R an equivalence relation? —
Yes, it is an equivalence relation.
17.
Let R be the relation on the set of points in a plane defined by (x1, y1) R (x2, y2) if x1 = x2. Is R an equivalence relation? —
No, it is not symmetric.
18.
Let R be the relation on the set of points in a plane defined by (x1, y1) R (x2, y2) if y1 = y2. Is R an equivalence relation? —
Yes, it is an equivalence relation.
19.
Let R be the relation on the set of points in a plane defined by (x1, y1) R (x2, y2) if x1 + y1 = x2 + y2. Is R an equivalence relation? —
Yes, it is an equivalence relation.
20.
Consider the relation R on the set of triangles in a plane, where T1 R T2 if T1 is similar to T2. Is R an equivalence relation? —
Yes, it is an equivalence relation.
21.
Consider the relation R on the set of triangles in a plane, where T1 R T2 if T1 is congruent to T2. Is R an equivalence relation? —
Yes, it is an equivalence relation.
22.
Let R be the relation on the set of employees in a company, where a R b if a and b are in the same department. Is R an equivalence relation? —
Yes, it is an equivalence relation.
23.
Let R be the relation on the set of positive integers N defined by a R b if a divides b. Is R an equivalence relation? —
No, it is not symmetric.
24.
Let R be the relation on the set of positive integers N defined by a R b if a = b. Is R an equivalence relation? —
Yes, it is an equivalence relation.
25.
Let R be the relation on the set of rational numbers Q defined by a R b if a - b is an integer. Is R an equivalence relation? —
Yes, it is an equivalence relation.
26.
Consider the relation R on the set of all strings of 'a's and 'b's, where s1 R s2 if s1 and s2 have the same number of 'a's. Is R an equivalence relation? —
Yes, it is an equivalence relation.
27.
Let A = {1, 2, 3, 4}. The relation R = {(a, b) | a and b are both even or a and b are both odd}. Is R an equivalence relation? —
Yes, it is an equivalence relation.
28.
Let R be the relation on the set of pairs of integers (x, y) such that x+y = x'+y' if (x, y) R (x', y'). Is R an equivalence relation? —
Yes, it is an equivalence relation.
29.
Let R be the relation on the set of pairs of integers (x, y) such that x*y = x'*y' if (x, y) R (x', y'). Is R an equivalence relation? —
No, it is not symmetric.
30.
Let A = {1, 2, 3, 4, 5, 6} and R be the relation a R b if a and b have the same remainder when divided by 3. Is R an equivalence relation? —
Yes, it is an equivalence relation.
31.
Consider the relation R on the set of all subsets of a given set S, where A R B if A ⊆ B. Is R an equivalence relation? —
No, it is not symmetric.
32.
Let R be the relation on the set of all people, where a R b if a and b are siblings. Is R an equivalence relation? —
Yes, it is an equivalence relation.
33.
Let R be the relation on the set of all people, where a R b if a and b are born in the same month. Is R an equivalence relation? —
Yes, it is an equivalence relation.
34.
Let R be the relation on the set of all people, where a R b if a and b have the same number of children. Is R an equivalence relation? —
Yes, it is an equivalence relation.
35.
Let A = {1, 2, 3}. Which of the following relations is an equivalence relation? —
R4 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}
36.
Let A = {1, 2, 3, 4} and R be the relation 'is equal to' on A. R = {(1, 1), (2, 2), (3, 3), (4, 4)}. Which property does R lack to be an equivalence relation? —
None, R is an equivalence relation.
37.
Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (a, b), (b, a)}. Which property is missing for R to be an equivalence relation? —
Transitive
38.
Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (a, b), (b, a), (a, c), (c, a)}. Which property is missing for R to be an equivalence relation? —
Transitive
39.
The relation 'is perpendicular to' on the set of all lines in a plane is: —
Symmetric
40.
Consider the relation R on the set of integers Z, where a R b if and only if a - b is divisible by 5. Which property does R NOT possess? —
None of these
41.
If a relation R is symmetric and transitive, it is necessarily: —
None of the above
42.
Which property must a relation R on a set A satisfy to be considered an equivalence relation? —
Reflexive, Symmetric, and Transitive
43.
Let A be a non-empty set. The universal relation A x A on A is: —
An equivalence relation
44.
The relation 'is parallel to' on the set of all lines in a plane is: —
An equivalence relation
45.
Which of the following is NOT a property required for a relation to be an equivalence relation? —
Irreflexivity
46.
If a relation R is reflexive and transitive, it is necessarily: —
None of the above
47.
If R is an equivalence relation on A, and a ∈ A, then a belongs to which equivalence class? —
The equivalence class of any element b such that a R b
48.
In the context of equivalence relations, the equivalence classes partition the set. This means that the union of all equivalence classes is the original set, and the intersection of any two distinct equivalence classes is: —
The empty set
49.
If a relation R is reflexive and symmetric, it is necessarily: —
None of the above
50.
The set of all equivalence classes of an equivalence relation R on a set A forms a: —
Partition of the set A