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) Elements of $R$: Identify all pairs $(x, y)$ such that $x + y = 6$, where $x$ is from the set $A = \{1, 2, 3\}$ and $y$ is from the set $B = \{4, 5, 6\}$. - For $x = 1$, $y = 6 - 1 = 5$. The pair is $(1, 5)$.
- For $x = 2$, $y = 6 - 2 = 4$. The pair is $(2, 4)$.
- For $x = 3$, $y = 6 - 3 = 3$. Since $3$ is not an element of $B$, no pair is formed. Therefore, the relation $R$ is
\[ R = \{(1, 5), (2, 4)\}. \]2. (ii) Is $R$ a function? Justify. A relation is a function if each element in the domain maps to exactly one element in the codomain. - $x = 1$ maps to $y = 5$, and $x = 2$ maps to $y = 4$. - No element in $A$ is mapped to multiple elements in $B$. Therefore, $R$ is a function.3. (iii) Domain and Range of $R$: - The domain of $R$ consists of all $x$ values from $A$ that are part of a pair in $R$. The domain is \[ \text{Domain}(R) = \{1, 2\}. \] - The range of $R$ consists of all $y$ values from $B$ that are paired with an $x$ from $A$. The range is \[ \text{Range}(R) = \{4, 5\}. \]