Is Person to Mother Relation a Function?
📺 Video Explanation
📝 Question
Consider the relation:
\[ \{(x,y): x \text{ is a person, } y \text{ is the mother of } x\} \]
Check:
- Is it a function?
- If yes, is it injective?
- Is it surjective?
✅ Solution
🔹 Is it a Function?
A relation is a function if every input has exactly one output.
Here:
- Each person has exactly one biological mother.
So every person maps to one unique mother.
✔ Yes, it is a function.
🔹 Is it Injective?
A function is injective if different inputs have different outputs.
But:
- Two siblings can have the same mother.
Example:
Rahul and Riya may both map to same mother.
❌ Not injective.
🔹 Is it Surjective?
For surjective:
every element in codomain must be image of some input.
If codomain is set of all mothers:
- Every mother is mother of at least one person.
✔ Onto (with codomain = set of mothers).
If codomain is all women/persons:
- Not every woman is mother.
So surjective depends on codomain.
🎯 Final Answer
✔ It is a function.
❌ Not injective.
✔ Surjective only if codomain = set of mothers.
🚀 Exam Shortcut
- Each person has one mother → function
- Siblings break injectivity
- Always check codomain before deciding onto