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

To evaluate the relations \( R \) defined on the sets \( X = \{2, 4, 6\} \) and \( Y = \{4, 8, 12, 16\} \), we will check each relation and determine if it defines a function. A relation is a function if every element in the domain (set \( X \)) maps to exactly one element in the codomain (set \( Y \)). ### 1. Relation \( R: (x+y) < y \) We need to check for each \( x \in X \) if there exists a \( y \in Y \) such that \( (x+y) < y \). - For \( x = 2 \): - \( 2 + y < y \) implies \( 2 < 0 \) (no valid \( y \)). - For \( x = 4 \): - \( 4 + y < y \) implies \( 4 < 0 \) (no valid \( y \)). - For \( x = 6 \): - \( 6 + y < y \) implies \( 6 < 0 \) (no valid \( y \)). **Conclusion**: There are no valid pairs \( (x, y) \) for this relation. Thus, the mapping \( d \) is empty, and it is not a function. ### 2. Relation \( R: 2x = y \) We will check for each \( x \in X \) if there exists a \( y \in Y \) such that \( 2x = y \). - For \( x = 2 \): - \( 2 \cdot 2 = 4 \) (valid, \( y = 4 \)). - For \( x = 4 \): - \( 2 \cdot 4 = 8 \) (valid, \( y = 8 \)). - For \( x = 6 \): - \( 2 \cdot 6 = 12 \) (valid, \( y = 12 \)). Now we can summarize the mapping: - \( 2 \mapsto 4 \) - \( 4 \mapsto 8 \) - \( 6 \mapsto 12 \) **Conclusion**: Each \( x \) in \( X \) maps to exactly one \( y \) in \( Y \). Thus, this relation is a function. ### 3. Relation \( R: x + 1 \) divides \( y - 1 \) with zero remainder We will check for each \( x \in X \) if there exists a \( y \in Y \) such that \( y - 1 \) is divisible by \( x + 1 \). - For \( x = 2 \): - \( x + 1 = 3 \) - Check \( y - 1 \) for \( y \in Y \): - \( 4 - 1 = 3 \) (divisible by 3) - \( 8 - 1 = 7 \) (not divisible) - \( 12 - 1 = 11 \) (not divisible) - \( 16 - 1 = 15 \) (not divisible) - Valid \( y = 4 \). - For \( x = 4 \): - \( x + 1 = 5 \) - Check \( y - 1 \): - \( 4 - 1 = 3 \) (not divisible) - \( 8 - 1 = 7 \) (not divisible) - \( 12 - 1 = 11 \) (not divisible) - \( 16 - 1 = 15 \) (not divisible) - No valid \( y \). - For \( x = 6 \): - \( x + 1 = 7 \) - Check \( y - 1 \): - \( 4 - 1 = 3 \) (not divisible) - \( 8 - 1 = 7 \) (divisible by 7) - \( 12 - 1 = 11 \) (not divisible) - \( 16 - 1 = 15 \) (not divisible) - Valid \( y = 8 \). Now we can summarize the mapping: - \( 2 \mapsto 4 \) - \( 6 \mapsto 8 \) **Conclusion**: The mapping is not defined for \( x = 4 \) (no valid \( y \)). Thus, this relation is not a function. ### Summary of Results 1. \(