Mathematics Question on Relations

Question

Let f(x) = {1,2,3,4,5,6,7}, the relation R ={(x,y) ϵ A x A, x + y = 7} is

Answer 1

Solution

Let's check the given relation R on the set A = {1,2,3,4,5,6,7} for the properties of symmetry, reflexivity, and transitivity:
Symmetric: If (x,y) is in R, then x + y = 7. But (y,x) is also in R, since y + x = x + y = 7. Therefore, the relation is symmetric.
Reflexive: For any element x in A, x + x = 2x. Since 2x is not equal to 7 for any x in A, (x,x) cannot be in R. Therefore, the relation is not reflexive.
Transitive: If (x,y) and (y,z) are in R, then x + y = 7 and y + z = 7. Adding these equations, we get x + y + y + z = 14, or x + 2y + z = 14. But this
equation does not necessarily imply that (x,z) is in R, since 2y may not equal 7 - x - z for some choices of x, y, and z in A. Therefore, the relation is not transitive.
Since the relation is symmetric, but not reflexive or transitive, it is not an equivalence relation.
Answer. A