Question

Let X= {a,b,c} and consider the relation R= {(a,a), (b,b), (c,c)}. Then, R is

• A. reflexive
• B. transitive
• C. symmetric
• D. All of the above

Answer 1

Answer: (D)

All of the above

Answer 2

Let the set be $X = \{a, b, c\}$ and the relation be $R = \{(a, a), (b, b), (c, c)\}$.

We need to check if the relation R is reflexive, symmetric, and transitive.

1. Reflexive: A relation R on a set X is reflexive if for every element $x \in X$, $(x, x) \in R$. In this case, $X = \{a, b, c\}$. We check if $(a, a) \in R$, $(b, b) \in R$, and $(c, c) \in R$. From the given relation $R = \{(a, a), (b, b), (c, c)\}$, we see that $(a, a) \in R$, $(b, b) \in R$, and $(c, c) \in R$. Thus, R is reflexive.

2. Symmetric: A relation R on a set X is symmetric if for every pair $(x, y) \in R$, the pair $(y, x)$ is also in R. We check each pair in R:

   • For $(a, a) \in R$, we check if $(a, a) \in R$. Yes, it is.
   • For $(b, b) \in R$, we check if $(b, b) \in R$. Yes, it is.
   • For $(c, c) \in R$, we check if $(c, c) \in R$. Yes, it is. There are no pairs $(x, y)$ in R where $x \neq y$. The condition "if $(x, y) \in R$, then $(y, x) \in R$" is vacuously true for all pairs where $x \neq y$. For pairs where $x=y$, the condition is "if $(x, x) \in R$, then $(x, x) \in R$", which is always true. Thus, R is symmetric.

3. Transitive: A relation R on a set X is transitive if for every $x, y, z \in X$, whenever $(x, y) \in R$ and $(y, z) \in R$, then $(x, z)$ must also be in R. We consider all possible combinations of pairs $(x, y)$ and $(y, z)$ that are both in R. The pairs in R are only of the form $(element, element)$. If $(x, y) \in R$, then $x$ must be equal to $y$. If $(y, z) \in R$, then $y$ must be equal to $z$. So, if $(x, y) \in R$ and $(y, z) \in R$, it implies $x=y$ and $y=z$, which means $x=y=z$. The condition for transitivity becomes: If $(x, x) \in R$ and $(x, x) \in R$, then $(x, x) \in R$. Since $(x, x)$ is in R for $x \in \{a, b, c\}$, this condition is satisfied. Thus, R is transitive.

Since the relation R is reflexive, symmetric, and transitive, it satisfies all three properties listed in options (a), (b), and (c). Therefore, option (d) is correct.