Relations

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.

To determine the nature of the relation , we need to evaluate whether it is reflexive, symmetric, or transitive. The relation is defined as .

Reflexive: A relation is reflexive if every element is related to itself. For lines , to be reflexive, must be perpendicular to itself. However, no line can be perpendicular to itself. Therefore, is not reflexive.

Symmetric: A relation is symmetric if whenever , then . If , then . Thus, if one line is perpendicular to another, the reverse is also true. Therefore, is symmetric.

Transitive: A relation is transitive if whenever and , then . If and , there is no guarantee that is perpendicular to . Therefore, is not transitive.

Therefore, the relation is symmetric but neither reflexive nor transitive.
Correct answer: B, C only.

To determine the properties of the relation R defined on the set of children where aRb if "a is a brother of b", we analyze its characteristics:

Therefore, the relation R is symmetric but not transitive. Hence, the statement "transitive but not symmetric" is incorrect based on conventional definitions of symmetry and transitivity, despite it being noted as the correct answer. Given the scenario, evaluating based on usual set relation properties suggests R should be symmetric.