Closures of Relations MCQ Quiz in தமிழ் - Objective Question with Answer for Closures of Relations - இலவச PDF ஐப் பதிவிறக்கவும்
Last updated on Apr 20, 2025
Latest Closures of Relations MCQ Objective Questions
Top Closures of Relations MCQ Objective Questions
Closures of Relations Question 1:
Consider R and S be two equivalence relations, which of the following is true regarding the R and S?
Answer (Detailed Solution Below)
Closures of Relations Question 1 Detailed Solution
Option D is wrong as transitivity may break on taking the union of two equivalence relations
For two equivalence relation R and S
- The largest equivalence relation in R and S is
- The smallest equivalence relation which contains R and S is
So option A is correct.
Option C is trivially false as option A is correct.
Closures of Relations Question 2:
Let R be a relation on the set of ordered pairs of positive integers such that ((p, q), (r, s)) ∈ R if and only if p – s = q – r. Which one of the following is true about R?
Answer (Detailed Solution Below)
Closures of Relations Question 2 Detailed Solution
Given: ((p, q), (r, s)) ∈ R
Only if p – s = q – r
Consider,
If ((p,q), (p,q)) ∈ R, then, p – q = q – p
It is not true for any positive integer p and q.
So, relation is not reflexive.
For symmetric: if ((p,q),(r,s)) ∈ R then also, ((r,s), (p,q)) ∈ R
Means if p – s = q – r then r – q = s – p
Both are equal, so relation is symmetric.Closures of Relations Question 3:
Consider the following statements:
(I). If
(II). If
(III). If
Which of the above statements is/are TRUE?
Answer (Detailed Solution Below)
Closures of Relations Question 3 Detailed Solution
(1).
(2). Inverse of an equivalence relation is also an equivalence relation.
(3) If
Hence, all three are true.
Closures of Relations Question 4:
What is the possible number of reflexive relations on a set of 5 elements?
Answer (Detailed Solution Below)
Closures of Relations Question 4 Detailed Solution
The correct answer is: Option 3) 220
Explanation:
Let the set be A = {a₁, a₂, a₃, a₄, a₅}. The number of elements in the set is 5.
A relation R on a set A is defined as a subset of A × A, i.e., all ordered pairs formed from elements of A.
The total number of elements in A × A is:
n × n = 5 × 5 = 25 ordered pairs.
Reflexive relation:
A relation is reflexive if every element is related to itself. So the relation must include:
- (a₁, a₁)
- (a₂, a₂)
- (a₃, a₃)
- (a₄, a₄)
- (a₅, a₅)
There are 5 such mandatory pairs that must be included in every reflexive relation.
The remaining 25 - 5 = 20 pairs are optional — they can be included or not.
So, for each of the 20 remaining pairs, we have 2 choices (include or exclude).
Thus, the total number of reflexive relations is:
220
✅ Hence, the correct answer is: Option 3) 220
Closures of Relations Question 5:
A relation 'R' is defined on ordered pairs of integers as:
(x, y) R (u, v) if x < u and y > v. Then R is
Answer (Detailed Solution Below)
Closures of Relations Question 5 Detailed Solution
The correct answer is Neither a partial order nor an equivalence relation
Key Points
Let's reevaluate the properties of the relation R : if and .
- Reflexivity: For any ordered pair , it is not possible for both and to be true simultaneously. Therefore, is not reflexive.
- Antisymmetry: If and , then and imply and . However, this does not necessarily mean that . Therefore, is not antisymmetric.
- Transitivity: If and , then , , , and . Combining these, we can deduce and . Therefore, is transitive.
- A binary relation is an equivalence relation on a nonempty set S if and only if the relation is reflexive(R), symmetric(S) and transitive(T).
- A binary relation is a partial order if and only if the relation is reflexive(R), antisymmetric(A) and transitive(T).
- From the given relation, it is neither partial order nor equivalence relation.
So, the correct answer is indeed: 1) Neither a partial order nor an equivalence relation.
Closures of Relations Question 6:
Which of the relations on {0, 1, 2, 3} is an equivalence relation?
Answer (Detailed Solution Below)
Closures of Relations Question 6 Detailed Solution
Concept:
Equivalence relation: A relation is said to be equivalence relation if it is reflexive, symmetric and transitive relation.
- Reflexive relation: a relation in which every element maps to itself.
- Symmetric relation: a relation is symmetric if there exists (a, b), then (b, a) should also be in relation.
- Transitive relation: A relation is transitive if there exists (a, b) and (b, c) then there must also exist (a, c) in the relation.
Explanation:
Given set = {0,1,2, 3}
(1) {(0, 0) (0, 2) (2, 0) (2, 2) (2, 3) (3, 2) (3, 3)}
It is not equivalence relation. Here, element 1 is not matched to itself. It is not reflexive.
(2) {(0, 0) (1, 1) (2, 2) (3, 3)}
It is equivalence relation. As, it is satisfying property of all three: reflexive, symmetric and transitive.
(3) {(0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0)}
This relation is neither reflexive nor symmetric so, it cannot be equivalence relation.
(4) {(0, 0) (0, 2) (2, 3) (1, 1) (2, 2)}
This is neither reflexive nor symmetric. So, it cannot be equivalence relation.
Closures of Relations Question 7:
Which of the relations on {0, 1, 2, 3} is an equivalence relation?
Answer (Detailed Solution Below)
Closures of Relations Question 7 Detailed Solution
Concept:
Equivalence relation: A relation is said to be equivalence relation if it is reflexive, symmetric and transitive relation.
- Reflexive relation: a relation in which every element maps to itself.
- Symmetric relation: a relation is symmetric if there exists (a, b), then (b, a) should also be in relation.
- Transitive relation: A relation is transitive if there exists (a, b) and (b, c) then there must also exist (a, c) in the relation.
Explanation:
Given set = {0,1,2, 3}
(1) {(0, 0) (0, 2) (2, 0) (2, 2) (2, 3) (3, 2) (3, 3)}
It is not equivalence relation. Here, element 1 is not matched to itself. It is not reflexive.
(2) {(0, 0) (1, 1) (2, 2) (3, 3)}
It is equivalence relation. As, it is satisfying property of all three: reflexive, symmetric and transitive.
(3) {(0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0)}
This relation is neither reflexive nor symmetric so, it cannot be equivalence relation.
(4) {(0, 0) (0, 2) (2, 3) (1, 1) (2, 2)}
This is neither reflexive nor symmetric. So, it cannot be equivalence relation.
Closures of Relations Question 8:
Which of the relations on {0, 1, 2, 3} is an equivalence relation?
Answer (Detailed Solution Below)
Closures of Relations Question 8 Detailed Solution
Concept:
Equivalence relation: A relation is said to be equivalence relation if it is reflexive, symmetric and transitive relation.
- Reflexive relation: a relation in which every element maps to itself.
- Symmetric relation: a relation is symmetric if there exists (a, b), then (b, a) should also be in relation.
- Transitive relation: A relation is transitive if there exists (a, b) and (b, c) then there must also exist (a, c) in the relation.
Explanation:
Given set = {0,1,2, 3}
(1) {(0, 0) (0, 2) (2, 0) (2, 2) (2, 3) (3, 2) (3, 3)}
It is not equivalence relation. Here, element 1 is not matched to itself. It is not reflexive.
(2) {(0, 0) (1, 1) (2, 2) (3, 3)}
It is equivalence relation. As, it is satisfying property of all three: reflexive, symmetric and transitive.
(3) {(0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0)}
This relation is neither reflexive nor symmetric so, it cannot be equivalence relation.
(4) {(0, 0) (0, 2) (2, 3) (1, 1) (2, 2)}
This is neither reflexive nor symmetric. So, it cannot be equivalence relation.
Closures of Relations Question 9:
Let S be a set of n elements. The number of ordered pairs in largest and smallest equivalence relation on set S are respectively.
Answer (Detailed Solution Below)
Closures of Relations Question 9 Detailed Solution
Equivalence Relations :
Let R be a relation on set A. If R is reflexive, symmetric, and transitive then it is said to be an equivalence relation. Consequently, two elements a and b related by an equivalence relation are said to be equivalent.
Consider a relation A (1,2,3)
The largest equivalence relation on A is A x A.
Largest ordered set are A x A = { (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) }
The number of elements in A x A =n2 elements.
Smallest ordered set are { (1,1) (2,2) ( 3,3)}
The smallest equivalence relation on A is the diagonal relation with n elements.
Hence the correct answer is n2 and n.
Closures of Relations Question 10:
A relation R is said to be circular if aRb and bRc together imply cRa. Which of the following options is/are correct?
Answer (Detailed Solution Below)
Closures of Relations Question 10 Detailed Solution
Answer: Option 3
Concept:
Equivalence Relation:
For a Relation to be Equivalence Relation, Relation has to satisfy 3 property ;
1. Reflexive Property
2. Symmetric property
3. Transitive Property
Explanation:
Let's start with Option 3;
Option 3:if a relation S is reflexive and circular, then S is an equivalence relation.
It is correct.
- S is reflexive and circular; see that if are able to derive Symmetric and transitive property from these given properties.
aRa ∈ S and aRb ∈ S => bRa ∈ S /Using Reflexive and circular
aRb and bRa both exist then the relation is Symmetric.
- Now
aRb and bRc together => cRa and according to symmetricity ≡ aRc
aRb and bRc together => aRc
Hence Transitivity satisfied.
Hence S is an Equivalence Relation.
Example : S = { {a,a) (b.b) (c,c) } (diagonal relation)
Option 1: If a relation S is transitive and circular, then S is an equivalence relation.
It is not correct.
Consider a example of empty relation or S = {(a,b)}; it is transitive and circular but its not Equivalence Relation.
Hence S is not an equivalence Relation.
Option 2: If a relation S is reflexive and symmetric, then S is an equivalence relation.
It is not correct. For a Relation S to be Equivalence Reflexive, Symmetric and Transitive are required.
Option 4: if a relation S is circular and symmetric, then S is an equivalence relation.
It is not correct.
Consider a example of empty relation or S = {(a,b) , (b,a), (a,a)}; It is circular and symmetric but its not Equivalence relation.
Hence S is not an equivalence Relation.