Relations and types of relations - One Line Questions
1.
A relation R on a set A is called symmetric if for every a, b ∈ A, whenever (a, b) ∈ R, then which of the following must also be true? —
(b, a) ∈ R
2.
Let A = {a, b, c}. Which relation is the identity relation on A? —
{(a,a), (b,b), (c,c)}
3.
If A = {1, 2} and B = {3, 4}, how many possible relations are there from A to B? —
16
4.
What is a binary relation on a set A? —
A relation from set A to set A
5.
What is an equivalence relation? —
A relation that is reflexive, symmetric, and transitive
6.
What is an irreflexive relation on a set A? —
A relation where for all a ∈ A, (a, a) ∉ R
7.
What is the definition of a relation R from a set A to a set B? —
A subset of the Cartesian product A x B
8.
If R is an equivalence relation on A, it partitions the set A into disjoint subsets called: —
Equivalence classes
9.
Which type of relation is reflexive, symmetric, and transitive? —
Equivalence relation
10.
Since the relation 'x - y is divisible by 5' on integers is reflexive, symmetric, and transitive, it is an: —
Equivalence relation
11.
If a relation R on set A is transitive, what condition must hold for any elements a, b, c ∈ A? —
If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R
12.
Let A = {1, 2, 3} and R = {(1,2), (2,1)}. Is R transitive on A? —
Yes, because there are no elements a, b, c such that (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R
13.
Let A = {1, 2, 3} and R = {(1,2), (2,1)}. Is R reflexive on A? —
No, because (1,1), (2,2), (3,3) are not in R
14.
Let A = {1, 2, 3} and R = {(1,2), (2,1)}. Is R symmetric on A? —
Yes, because if (1,2) ∈ R, then (2,1) ∈ R
15.
Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,1)}. Is R transitive on A? —
No, because (1,2) ∈ R and (2,1) ∈ R, but (1,1) ∈ R. Let's check other combinations. (1,2) ∈ R and (2,2) ∈ R, implies (1,2) ∈ R. (2,1) ∈ R and (1,1) ∈ R, implies (2,1) ∈ R. The issue arises from (1,2) and (2,1).
16.
Let A = {1, 2, 3, 4} and R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1)}. Is R transitive? —
No, because (1,2) ∈ R and (2,1) ∈ R, but (1,1) ∈ R. This condition is met. However, if we consider (2,1) ∈ R and (1,2) ∈ R, then (2,2) ∈ R. Let's recheck.
17.
Let A = {1, 2} and R = {(1,1), (1,2), (2,1)}. Is R transitive on A? —
No, because (1,2) ∈ R and (2,1) ∈ R, but (1,1) is present. However, consider (1,2) and (2,1). We have (1,1) ∈ R. Now consider (2,1) and (1,1). We need (2,1) ∈ R, which is true. The issue is if we have (a,b) and (b,c) implies (a,c). Here, (1,2) ∈ R and (2,1) ∈ R, so we must have (1,1) ∈ R, which is true. Let's re-evaluate.
18.
Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3), (1,2), (2,3)}. Is R transitive on A? —
No, because (1,2) ∈ R and (2,3) ∈ R, but (1,3) ∉ R
19.
Let A = {1, 2, 3} and R = {(1,2), (2,3), (3,1)}. Is R transitive? —
No, because (1,2) ∈ R and (2,3) ∈ R, but (1,3) ∉ R
20.
Let A = {1, 2, 3, 4} and R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2)}. Is R transitive? —
No, because (1,2) ∈ R and (2,3) ∈ R, but (1,3) ∉ R
21.
Let A = {1, 2} and R = {(1,1), (1,2), (2,1)}. Is R reflexive on A? —
No, because (2,2) ∉ R
22.
Consider the relation R on the set of integers defined by x R y if x > y. Is this relation symmetric? —
No, because if x > y, then y > x is false (e.g., 5 > 3 but 3 is not > 5)
23.
Consider the relation R on the set of real numbers defined by x R y if x ≤ y. Is this relation symmetric? —
No, because x ≤ y does not imply y ≤ x (e.g., 2 ≤ 3 but 3 is not ≤ 2)
24.
Let R be a relation on set A. If R is symmetric and transitive, does it imply R is reflexive? —
No, not necessarily. For example, R = {(1,2), (2,1)} on A = {1, 2, 3}.
25.
Let R be a relation on a set A. Which of the following is NOT a necessary condition for R to be an equivalence relation? —
R must contain at least one non-diagonal pair (a,b) where a != b
26.
Which type of relation is defined as a relation R on a set A such that for every element a ∈ A, (a, a) ∈ R? —
Reflexive relation
27.
Let A = {1, 2, 3} and R = {(1,1), (2,2), (3,3)}. Which type of relation is R on A? —
Reflexive, Symmetric, and Transitive
28.
The relation 'is perpendicular to' on the set of all lines in a plane is: —
Symmetric but not reflexive or transitive
29.
If R is a relation on set A, what does the notation 'a R b' signify? —
The ordered pair (a, b) is in the relation R
30.
What is the empty relation on a set A? —
The relation ∅
31.
What is the universal relation on a set A? —
The relation A x A
32.
If a relation R on set A is not transitive, what does it mean? —
There exist elements a, b, c ∈ A such that (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R
33.
If a relation R on set A is not reflexive, what does it mean? —
There exists at least one element a ∈ A such that (a, a) ∉ R
34.
If a relation R on set A is not symmetric, what does it mean? —
There exists at least one pair (a, b) ∈ R such that (b, a) ∉ R
35.
What property is shared by both the empty relation and the universal relation on a non-empty set? —
Transitivity
36.
Which of the following relations on the set of natural numbers N is NOT an equivalence relation? —
x R y if x divides y
37.
Let A = {1, 2} and R = {(1,1), (1,2), (2,1)}. Is R symmetric on A? —
Yes, because (1,2) ∈ R implies (2,1) ∈ R and (1,1) ∈ R implies (1,1) ∈ R
38.
Consider the relation R on the set of all lines in a plane defined by L1 R L2 if L1 is parallel to L2. Is this relation reflexive? —
Yes, because every line is parallel to itself
39.
Consider the relation R on the set of all lines in a plane defined by L1 R L2 if L1 is parallel to L2. Is this relation transitive? —
Yes, because if L1 is parallel to L2 and L2 is parallel to L3, then L1 is parallel to L3
40.
Consider the relation R on the set of all lines in a plane defined by L1 R L2 if L1 is parallel to L2. Is this relation symmetric? —
Yes, because if L1 is parallel to L2, then L2 is parallel to L1
41.
Consider the relation R on the set Z of integers defined by x R y if x - y is divisible by 5. Is R transitive? —
Yes, because if x - y = 5k and y - z = 5m (k, m integers), then x - z = (x - y) + (y - z) = 5k + 5m = 5(k + m), which is divisible by 5
42.
Consider the relation R on the set Z of integers defined by x R y if x - y is divisible by 5. Is R symmetric? —
Yes, because if x - y = 5k for some integer k, then y - x = -5k = 5(-k), which is also divisible by 5
43.
Consider the relation R on the set of integers defined by x R y if x > y. Is this relation transitive? —
Yes, because if x > y and y > z, then x > z
44.
Consider the relation R on the set of real numbers defined by x R y if x ≤ y. Is this relation transitive? —
Yes, because if x ≤ y and y ≤ z, then x ≤ z
45.
Consider the relation R on the set Z of integers defined by x R y if x - y is divisible by 5. Is R reflexive? —
Yes, because x - x = 0, which is divisible by 5
46.
Consider the relation R on the set of integers defined by x R y if x > y. Is this relation irreflexive? —
Yes, because x > x is never true for any integer x
47.
Consider the relation R on the set of real numbers defined by x R y if x ≤ y. Is this relation reflexive? —
Yes, because x ≤ x for all real numbers x