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 :
(C) and (E) Only
A relation R on a set A is reflexive when (a, a) ∈ R for every a ∈ A. For A = {1, 2, 3, 4} the relation must contain all four pairs (1, 1), (2, 2), (3, 3) and (4, 4).
Step 1 — R1 contains (1, 1), (2, 2), (4, 4) but not (3, 3), so it is not reflexive.
Step 2 — R2 contains only (1, 1) among the diagonal pairs; (2, 2), (3, 3), (4, 4) are missing, so it is not reflexive.
Step 3 — R3 contains (1, 1), (2, 2), (3, 3) and (4, 4), so it is reflexive.
Step 4 — R4 contains no diagonal pair at all; it is irreflexive.
Step 5 — R5 contains (1, 1), (2, 2), (3, 3) and (4, 4), so it is reflexive (it is in fact the relation ≤).
Only R3 and R5 are reflexive; choices including R1, R2 or R4 are wrong.
(C) and (E) Only option is correct.
Similar Questions
The relation R={(1,3), (1,1), (3,1), (1,2), (3,3), (4,4)} defined in {1,2,3,4} is ______.
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 \(R \leq S,then \overline{S} \leq \overline{R}\)
