Question 7978 – GATE 2017 [Set-2]November 16, 2023
Question 10288 – Set-TheoryNovember 16, 2023
A relation R is said to be circular of aRb and bRc together imply cRa. Which of the following options is/are correct?
Correct Answer: A
Theorem: A relation R on a set A is an equivalence relation if and only if it is reflexive and circular.
For symmetry, assume that x, y ∈ A so that xRy, lets check for yRx.
Since R is reflexive and y ∈ A, we know that yRy. Since R is circular and xRy and yRy, we know that yRx. Thus R is symmetric.
For transitivity, assume that x, y, z ∈ A so that xRy and yRz. Check for xRz. Since R is circular and xRy and yRz, we know that zRx. Since we already proved that R is symmetric, zRx implies that xRz. Thus R is transitive.