Interesting Problem from Gelfand's Algebra; Relevance?

[Axel Harper]

Homework Statement

Problem 63 from Gelfand's book Algebra asks "are the father of the son of NN and the son of the father of NN the same person?"

Homework Equations

This problem is in a section about the square of a sum formula.
(a+b)² = a²+2ab+b²

The Attempt at a Solution

If NN has a biological son x, then x's biological father must be NN. If NN has a biological father y, then y's biological son is not necessarily NN because NN could have brothers.
When I first did this problem a couple years ago I wondered how this was relevant at all. Now I interpret this as Gelfand's way of introducing the idea that both a² and (-a)² equal a². Rational exponents are covered already in an earlier section. Can anybody confirm this, or does anyone have a differing interpretation?

 

[Mogarrr]

If NN has a biological son x, then x's biological father must be NN. If NN has a biological father y, then y's biological son is not necessarily NN because NN could have brothers.

I would think you're right. If you are, the answer would be "not always".

 

[thelema418]

The Attempt at a Solution

If NN has a biological son x, then x's biological father must be NN. If NN has a biological father y, then y's biological son is not necessarily NN because NN could have brothers.
When I first did this problem a couple years ago I wondered how this was relevant at all. Now I interpret this as Gelfand's way of introducing the idea that both a² and (-a)² equal a². Rational exponents are covered already in an earlier section. Can anybody confirm this, or does anyone have a differing interpretation?

Why does NN have to be male?

My thought was that the purpose of the problem is to encourage thinking about variables. In math history there were problems with solving $x^2 = 4$. Mathematicians would avoid a negative solution, such as $x=-2$. Similar issues arose with imaginary numbers.

I thought NN could be a woman. We may bias our interpretation of a variable if we impose a restriction.

 

[SammyS]

Why does NN have to be male?
...

Excellent point !

 

[Axel Harper]

Why does NN have to be male?

That's a good point. I think we could still interpret the problem in the same manner if NN is a woman because we still can't guarantee that her son's father is the same person as her father's son.