V. For \( \mathrm{X}=\{2,4,6\} \) and \( \mathrm{Y}=\{4,8,12,16\} \), evaluate the following relation by illustrating its mapping diagram and stating if it is a function or not a function .No solution means zero. 1. \( R:(\mathrm{x}+\mathrm{y})<\mathrm{y} \) 2. \( R: 2 \mathrm{x}=\mathrm{y} \) 3. \( \mathrm{R}: \mathrm{x}+1 \) divides \( \mathrm{y}-1 \) with zero remainder.

To evaluate the relations given the sets \( \mathrm{X} = \{2, 4, 6\} \) and \( \mathrm{Y} = \{4, 8, 12, 16\} \), we will analyze each relation and determine if it is a function. A relation is a function if every element in the domain (set \( \mathrm{X} \)) maps to exactly one element in the codomain (set \( \mathrm{Y} \)). ### 1. Relation \( R: (x + y) < y \) We can rewrite the inequality as \( x < 0 \). Since all elements in \( \mathrm{X} \) are positive (2, 4, 6), there are no pairs \( (x, y) \) that satisfy this condition. Therefore, the relation is empty. **Mapping Diagram:** - No arrows since there are no valid pairs. **Conclusion:** This relation is **not a function** (it is an empty relation). --- ### 2. Relation \( R: 2x = y \) We will check each \( x \) in \( \mathrm{X} \) to see if it produces a corresponding \( y \) in \( \mathrm{Y} \): - For \( x = 2 \): \( 2(2) = 4 \) (valid, \( 4 \in \mathrm{Y} \)) - For \( x = 4 \): \( 2(4) = 8 \) (valid, \( 8 \in \mathrm{Y} \)) - For \( x = 6 \): \( 2(6) = 12 \) (valid, \( 12 \in \mathrm{Y} \)) **Mapping Diagram:** - \( 2 \rightarrow 4 \) - \( 4 \rightarrow 8 \) - \( 6 \rightarrow 12 \) **Conclusion:** This relation is a **function** (each \( x \) maps to exactly one \( y \)). --- ### 3. Relation \( R: x + 1 \) divides \( y - 1 \) with zero remainder We will check each \( x \) in \( \mathrm{X} \) and see if \( x + 1 \) divides \( y - 1 \) for each \( y \) in \( \mathrm{Y} \): - For \( x = 2 \): \( x + 1 = 3 \) - \( y = 4 \): \( 4 - 1 = 3 \) (divisible) - \( y = 8 \): \( 8 - 1 = 7 \) (not divisible) - \( y = 12 \): \( 12 - 1 = 11 \) (not divisible) - \( y = 16 \): \( 16 - 1 = 15 \) (not divisible) Valid pair: \( (2, 4) \) - For \( x = 4 \): \( x + 1 = 5 \) - \( y = 4 \): \( 4 - 1 = 3 \) (not divisible) - \( y = 8 \): \( 8 - 1 = 7 \) (not divisible) - \( y = 12 \): \( 12 - 1 = 11 \) (not divisible) - \( y = 16 \): \( 16 - 1 = 15 \) (not divisible) No valid pairs. - For \( x = 6 \): \( x + 1 = 7 \) - \( y = 4 \): \( 4 - 1 = 3 \) (not divisible) - \( y = 8 \): \( 8 - 1 = 7 \) (divisible) - \( y = 12 \): \( 12 - 1 = 11 \) (not divisible) - \( y = 16 \): \( 16 - 1 = 15 \) (not divisible) Valid pair: \( (6, 8) \) **Mapping Diagram:** - \( 2 \rightarrow 4 \) - \( 6 \rightarrow 8 \) **Conclusion:** This relation is **not a function** (since \( x = 2 \) maps to \( y = 4 \) and \( x = 6 \) maps to \( y = 8 \), but \( x = 4 \) has no mapping). --- ### Summary of Relations: 1. \( R: (x + y) < y \) - **Not a function** 2. \( R