Question

Let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$. A relation $R$ from $A$ to $B$ is defined as $R = \{(x, y) : x + y = 6, x \in A, y \in B \}$. (i) Write all elements of $R$.
(ii) Is $R$ a function? Justify.
(iii) Determine domain and range of $R$.

Hint:

A relation is a function if each element in the domain is related to exactly one element in the range.

Answer 1

1. (i) All elements of $R$: We need to find all pairs $(x, y)$ such that $x + y = 6$ with $x \in A = \{1, 2, 3\}$ and $y \in B = \{4, 5, 6\}$. - For $x = 1$, we have $y = 6 - 1 = 5$. So, the pair is $(1, 5)$.
- For $x = 2$, we have $y = 6 - 2 = 4$. So, the pair is $(2, 4)$.
- For $x = 3$, we have $y = 6 - 3 = 3$, but since $3 \notin B$, no pair is formed for $x = 3$. Therefore, the relation $R$ is
\[ R = \{(1, 5), (2, 4)\}. \] 2. (ii) Is $R$ a function? Justify. A relation is a function if every element of the domain is associated with exactly one element of the codomain. - Here, $x = 1$ is associated with $y = 5$ and $x = 2$ is associated with $y = 4$. - No element in $A$ is associated with more than one element in $B$. Therefore, $R$ is a function. 3. (iii) Domain and Range of $R$: - The domain of $R$ is the set of all $x$ values in $A$ for which there is a corresponding $y$ in $B$. In this case, the domain is \[ \text{Domain}(R) = \{1, 2\}. \] - The range of $R$ is the set of all $y$ values in $B$ that are associated with some $x$ in $A$. In this case, the range is \[ \text{Range}(R) = \{4, 5\}. \]