Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements
(S1): The number of elements in R is 18, and
(S2): The relation R is symmetric but neither reflexive nor transitive
Correct Answer:
C
Explanation
To evaluate the relation $R$ on the set $A = {0, 1, 2, 3, 4, 5}$, we first need to understand the conditions for an element $(x, y)$ to be in $R$. Specifically, $(x, y) ∈ R$ if and only if $max{x,y} ∈ {3,4}$.
Considering this, let's list the pairs:
For $max{x, y} = 3$, the possible pairs are:
$(0, 3), (3, 0), (1, 3), (3, 1), (2, 3), (3, 2), (3, 3)$
For $max{x, y} = 4$, the possible pairs are:
$(0, 4), (4, 0), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4)$
Combining these, the set $R$ consists of the following elements:
$R = {(0, 3), (3, 0), (1, 3), (3, 1), (2, 3), (3, 2), (3, 3), (0, 4), (4, 0), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4)}$
This gives us a total of 16 elements in $R$, not 18 as initially claimed in statement $S1$.
Next, we analyze the properties of the relation $R$:
Reflexivity: A relation is reflexive if $(x, x) ∈ R$ for all $x ∈ A$. For example, $(0, 0), (1, 1), (2, 2)$ are not in $R$, so $R$ is not reflexive.
Symmetry: A relation is symmetric if whenever $(a, b) ∈ R$, then $(b, a) ∈ R$ as well. For all pairs $(x, y)$ listed, both $(x, y)$ and $(y, x)$ are present. Thus, $R$ is symmetric.
Transitivity: A relation is transitive if whenever $(a, b) ∈ R$ and $(b, c) ∈ R$, then $(a, c) ∈ R$. An example where transitivity fails is $(0, 3)$ and $(3, 1)$ are in $R$ but $(0, 1)$ is not in $R$. Therefore, $R$ is not transitive.
In conclusion, statement $S2$ is correct as $R$ is symmetric but neither reflexive nor transitive.