Let A be the set of the children in a family. The relation ‘x is a brother of y' relation on A is

Options

A. reflexive
B. anti-symmetric
C. transitive
D. equivalence

Answer

To determine the type of relation defined by "x is a brother of y" on the set A (the set of children in a family), we will analyze the properties of the relation step by step.

### Step 1: Check for Reflexivity
A relation R is reflexive if, for every element x in A, the pair (x, x) is in R. This means that every element must be related to itself.

- In our case, if x is a child in the family, the statement "x is a brother of x" is not true. A person cannot be a brother of themselves.
- Therefore, (x, x) is not in R for any x in A.

**Conclusion**: The relation is **not reflexive**.

### Step 2: Check for Symmetry
A relation R is symmetric if, whenever (x, y) is in R, then (y, x) is also in R. This means if x is a brother of y, then y must also be a brother of x.

- If x is a brother of y, it implies that x is male. However, if y is female, then y cannot be a brother of x.
- Thus, if (x, y) is in R, (y, x) may not be in R.

**Conclusion**: The relation is **not symmetric**.

### Step 3: Check for Transitivity
A relation R is transitive if, whenever (x, y) is in R and (y, z) is in R, then (x, z) must also be in R. This means if x is a brother of y and y is a brother of z, then x must also be a brother of z.

- If x is a brother of y (x is male), and y is a brother of z (y is also male), then x can be a brother of z.
- Therefore, (x, z) is in R.

**Conclusion**: The relation is **transitive**.

### Step 4: Check for Anti-symmetry
A relation R is anti-symmetric if, whenever (x, y) and (y, x) are both in R, then x must be equal to y.

- In our case, if x is a brother of y and y is a brother of x, it does not imply that x = y (they can be different brothers).
- Thus, the relation does not satisfy the condition for anti-symmetry.

**Conclusion**: The relation is **not anti-symmetric**.

### Final Conclusion
Based on the analysis:
- The relation "x is a brother of y" is **not reflexive**, **not symmetric**, **transitive**, and **not anti-symmetric**.

Thus, the relation is classified as **transitive**.

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