Questions: Relation 3 (a, a),(g, x),(w, a),(x, a) Function Not a function Relation 4 (4, y),(2, z),(-9, y),(2, y) Function Not a function

Solution Steps

To determine if a relation is a function, we need to check if each input (first element of the pairs) maps to exactly one output (second element of the pairs). If any input maps to more than one output, it is not a function.

1. For each relation, create a dictionary to store the mappings.
2. Iterate through each pair in the relation.
3. Check if the input already exists in the dictionary with a different output.
4. If it does, the relation is not a function. Otherwise, it is a function.

The relation \( R_3 = \{(a, a), (g, x), (w, a), (x, a)\} \) consists of the pairs where the first element represents the input and the second element represents the output. To determine if \( R_3 \) is a function, we check if each input maps to exactly one output.

• The input \( a \) maps to \( a \).
• The input \( g \) maps to \( x \).
• The input \( w \) maps to \( a \).
• The input \( x \) maps to \( a \).

Since each input has a unique output, \( R_3 \) is a function.

The relation \( R_4 = \{(4, y), (2, z), (-9, y), (2, y)\} \) is analyzed similarly.

• The input \( 4 \) maps to \( y \).
• The input \( 2 \) maps to both \( z \) and \( y \).
• The input \( -9 \) maps to \( y \).

Since the input \( 2 \) maps to two different outputs (\( z \) and \( y \)), \( R_4 \) is not a function.

Final Answer