The set S has the property that if a is in S, then a squared + a is also in S. What set could be S? The answer is {1,0}. How do you get this answer?

What numbers squared are itself?

one and zero

But what exactly is a set and what is a?

I'm not sure, because the way you've stated the problem S should have more elements than just 1 and 0. You said
"...if a is in S, then a squared + a is also in S."
If I understand this correctly, then a is in S => a2 + a = a(a+1) is in S.
However, there are no restrictions placed on a, so it could be integers, rationals or real numbers.
You wrote, "The answer is {1,0}," but I don't see any rational for why those are the only possible numbers that can be in S. What I conclude is that if 0 is in S then 1 is in S, but if 1 is in S then 1*2=2 is in S, which means 2*3 is in S... Which means S is the triangular numbers, but this is only true if 0 is in S to begin with. Is there something you're omitting from the original question?

To find the set S that satisfies the given property, we can reason as follows:

1. We start by considering an arbitrary number a that belongs to S.
2. According to the property, if a is in S, then a squared + a is also in S.
3. Mathematically, this can be expressed as a^2 + a.
4. To find the numbers for which this expression is equal to a, we solve the equation a^2 + a = a.
5. This simplifies to a^2 = 0.
6. The only number that satisfies this equation is 0.
7. Therefore, 0 must be an element of S.

Now, let's consider the case when a = 0:

1. If we substitute a = 0 into the property equation, we get 0^2 + 0 = 0.
2. This shows that if 0 is in S, then 0 squared + 0 is also in S.
3. Therefore, 0 squared + 0 = 0 satisfies the property.

Combining both cases, we have established that 0 and 1 satisfy the given property. Thus, the set S could be {0, 1}.

However, it's important to note that the original question does not specify any restrictions on the type of numbers in S. Therefore, depending on the context, there could be other sets that also satisfy the given property.