The relation R={(1,3), (1,1), (3,1), (1,2), (3,3), (4,4)} defined in {1,2,3,4} is ______.
Not reflexive, not symmetric and not transitive
A relation R on A is reflexive if (a,a) ∈ R for every a ∈ A, symmetric if (a,b) ∈ R implies (b,a) ∈ R, anti-symmetric if (a,b) and (b,a) both in R force a = b, and transitive if (a,b), (b,c) ∈ R imply (a,c) ∈ R.
Step 1 — Reflexive? A = {1,2,3,4}; (1,1), (3,3), (4,4) are present but (2,2) is missing, so R is not reflexive.
Step 2 — Symmetric? (1,2) ∈ R but (2,1) ∉ R, so R is not symmetric. (It is not anti-symmetric either, because (1,3) and (3,1) are both present with 1 ≠ 3, and it is not asymmetric since it contains (1,1).)
Step 3 — Transitive? (3,1) ∈ R and (1,2) ∈ R, but (3,2) ∉ R, so R is not transitive.
The descriptions claiming reflexive, symmetric, anti-symmetric or asymmetric each contradict one of the checks above.
Not reflexive, not symmetric and not transitive option is correct.
Similar Questions
Consider the following relations on {1, 2, 3, 4}. Which of the following relations are reflexive ?
(A) R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}
(B) R2 = {(1, 1), (1, 2), (2, 1)}
(C) R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}
(D) R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}
(E) R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
Choose the correct answer from the options given below :
Important Questions from Relations
In the following question, select the related letters from the given alternatives.
MOB : ORG :: GRP : ?
The greatest integer function f : R → R given by f(x) = [x], (where [x] denotes the greatest integer), is _______
Set P has 4 elements and set Q has 5 elements. How many numbers of injections are defined from P to Q?
What is the scope of the definition of exponential function?
If R and S are relatives from A to B, then which of the following statements are false?
(i) If R ≤ S, then \( S^{-1}\leq R^{-1} \)
(ii) If \(R \leq S,then \overline{S} \leq \overline{R}\)
