Question

Can the following be a function:
\left\\{ \left( a,b \right):\text{ a is a person, b is an ancestor of a} \right\\}
If yes enter 1 else 0.

Answer 1

Hint: To solve the above question, we will have to make use of the fact that we only get a single value of a function when a variable is put in the function.

Complete step-by-step answer:
In the question, we are given that a is a person whose ancestor is b. In the question, it is not given who is the ancestor. Practically, the ancestor can be father, mother, grandfather, great grandfather, great grandmother and so on. So for different ancestor, we are going to take up different cases:
Case 1: b is father. Now the ordered pair (a , b) becomes ( person , father ). Thus the function we will get f (person, father). Let this function give value ${{x}_{1}}$ .
Case 2: b is mother. Now the ordered pair ( a , b ) becomes ( person , mother ). Thus the function we will get f (person, mother). The value of this function is assumed to be ${{x}_{2}}$ i.e. f (person, mother) = ${{x}_{2}}$ .
Case 3: b is grandfather. Now the order pair (a, b) becomes (person, grandfather). Thus the function we will get f (person, grandfather). The value of this function is assumed to be ${{x}_{3}}$ i.e. f (person, grandfather) =${{x}_{3}}$ .
Case IV: b is grandmother. Now the order pair (a , b) becomes (person , grandmother). Thus the function we will get f (person, grandmother). The value of this function is assumed to be ${{x}_{4}}$ i.e. f (person, grandmother) = ${{x}_{4}}$ .
Similarly we can form any number of cases. Thus from the above cases we get:
f (person, father) =${{x}_{1}}$ .
f (person, mother) =${{x}_{2}}$ .
f (person, grandfather) =${{x}_{3}}$ .
f (person, grandmother) =${{x}_{4}}$ .
As the mother, father, grandfather, grandmother all are ancestors, we get:
f (person, ancestor) = f (a, b) =${{x}_{1}}$ .
f (person, ancestor) = f (a, b) =${{x}_{2}}$ .
f (person, ancestor) = f (a, b) =${{x}_{3}}$ .
f (person, ancestor) = f (a, b) =${{x}_{4}}$ .
From above, we have got four different values of function at fixed points a and b. But according to the definition of function, there is only one value for the variables a and b. So our assumption was wrong. f (a, b) is not a function.
So the answer is 0.

Note: It is not possible for a function to have multiple values for the same variables but it is possible that the function may give some value for multiple variables. In other words, a function can have only a single image but it can have more than one pre – image. In our question, if it was given that who is the ancestor (like father, mother, etc) then f (a, b) could have been a function.