1. ## Write with symbols

(1) Equation $f(x) = g(x)$ has at most one real (R) solution.

$\exists ! x \in \mathbb{R} . f(x) = g(x)$

(2) Equation $f(x) = g(x)$ has more than one real (R) solution.

$\exists x \in \mathbb{R} . f(x) = g(x)$

(3) Equation $f(x) = g(x)$ has 2 real (R) solutions.

no clue for (3)

I know I did (1) wrong.. at most 1 means 1 or 0.. i wrote for 1.

(2) doesn't tell how mayn solutions there are; there could only be 1.

2. (1) Equation has at most one real (R) solution.
Perhaps surprisingly, this statement does not claim that there is a solution. Indeed, "at most one" includes, in particular, zero solutions. The sentence says that if there is a solution, then there is no other solution (i.e., different from the first). Equivalently, if there are two solutions, then they are equal. Still another way to say this, it is not the case that there exist two different solutions.

(2) Equation has more than one real (R) solution.
Your formula says that there exists a solution; it does not say that there exist more than one. A couple of ways to express this are:

There exist two different solutions.

It is not the case that A, where A is the formula from point (1).

(3) Equation has 2 real (R) solutions.
This is a little ambiguous. If one wants to be a literalist, this means that there exist two different solutions, maybe more; i.e., this is the same as (2). A more natural way is to say that there exists exactly two different solutions, i.e., there exist two different solutions and any solution is one of those two.

Explanation concerning "exactly two". Suppose a man comes to a friend and asks to borrow $100 for a week until the end of the month. The friend asks him how much money he has now, and he answers that he has$10. So the friend lends him a hundred bucks. However, if the friend knew that the man has $2000 available in a checking account besides$10 in cash and wants \$100 for a night in a casino, their friendship would probably be in danger.