So, the question I got was:
∀x,(x =/= 0) → (∃!y, xy = 1)
I'm really confused about what the "∃!y" means in this context, because the way I'm interpreting the question is:
"For all x (when x is not equal to 0), there does not exist a y such that x*y = 1." ???
But the answer is:
T (given x =/= 0, y = 1/x is the only example)
What am I not getting?
In the statement ∀x, (x =/= 0) → (∃!y, xy = 1), the symbol "∃!" represents the existential quantifier with uniqueness. It means "there exists one and only one."
Let's break down the statement and its meaning:
1. ∀x: This part states that the statement applies to all values of x.
2. (x =/= 0): This condition specifies that x is not equal to zero.
3. →: The arrow symbol "→" represents the implication or "if-then" relationship.
4. (∃!y, xy = 1): This part states that there exists one and only one y such that xy = 1.
Now, let's discuss your confusion about the interpretation and the correct answer.
Your interpretation, "For all x (when x is not equal to 0), there does not exist a y such that x*y = 1" is incorrect. The symbol "∃!" indicates that there indeed exists one and only one y that satisfies the equation xy = 1. Therefore, your interpretation contradicts the actual statement.
The correct interpretation is as follows: "For all x (when x is not equal to 0), there exists a unique y such that xy = 1."
To understand why the answer is T (true), let's consider an example. Suppose we take x = 2. According to the statement, there exists a unique y such that 2y = 1. Solving this equation, we find y = 1/2.
Similarly, for any other non-zero value of x, there will always exist a unique y such that xy = 1. For example, if x = 3, y would be 1/3, and so on.
Thus, the answer is T, meaning the statement is true. The existence of a unique y for every non-zero value of x is confirmed by finding that y = 1/x, which satisfies the equation xy = 1.