Let $A = \{-4, -3, -2, 0, 1, 3, 4\}$ and $R = \{(a, b) \in A \times A\}$ : $b = |a|$ or $b^{2} = a + 1$ be a…
Let $A = \{-4, -3, -2, 0, 1, 3, 4\}$ and $R = \{(a, b) \in A \times A\}$ : $b = |a|$ or $b^{2} = a + 1$ be a relation on $A$. Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is
Solution
$A = \{-4, -3, -2, 0, 1, 3, 4\}$ and $R = \{(a, b) \in A \times A : b = |a|$ or $b^{2} = a + 1\}$ be a relation on $A$.
So, the relation is given by,
$R = \{(-4, 4), (-3, 3), (0, 0), (1, 1), (3, 3), (4, 4), (0, 1), (3, -2)\}$
Now, relation to be reflexive $(a, a) \in R$ for all $a \in A$
$\Rightarrow (-4, -4), (-3, -3), (-2, -2)$ also should be added in $R$.
Now relation to be symmetric if $(a, b) \in R$, then $(b, a) \in R$ for all $a, b \in A$
$\Rightarrow (4, -4), (3, -3), (1, 0), (-2, 3)$ also should be added in $R$
Hence, minimum number of elements to be added to $R = 3 + 4 = 7$