Question

Given relation R={(x, y): y=x+5, x < 4, x, y ∈ N}. Where N is a set of natural numbers then :

• A. R is an equivalence relation.
• B. R is transitive but neither reflexive nor symmetric.
• C. R is reflexive but neither symmetric nor transitive.
• D. R is symmetric & transitive but not reflexive.

Answer 1

The Correct Option is B

The relation R is defined as R={(x, y): y=x+5, x<4, x, y ∈ N}. To determine the properties of this relation, let's examine each characteristic:

1. Reflexivity: A relation R on a set is reflexive if every element is related to itself. For R to be reflexive, (x, x) should be in R for every x. Since y=x+5, there's no natural number y such that y=x, hence R is not reflexive.

2. Symmetry: A relation is symmetric if whenever (x, y) is in R, (y, x) is also in R. If (x, y) ∈ R, then y=x+5. For symmetry, (y, x) should also satisfy y=x+5, implying x=y+5, which contradicts our initial definition unless x=y. So, R is not symmetric.

3. Transitivity: A relation is transitive if whenever (x, y) ∈ R and (y, z) ∈ R, (x, z) ∈ R. If (x, y) ∈ R, then y=x+5. For another pair with (y, z) ∈ R, z=y+5. Combining both, z=(x+5)+5=x+10. As (x, z) = (x, x+10) satisfies the condition for natural numbers under provided x conditions, R is transitive.

Hence, R is transitive but neither reflexive nor symmetric.