We define a new quantifier, uniqueness quantifier, the symbol of which is $\exists!.$
For any predicate $\text{P}$ and universe $\text{U}, \exists! x \text{P}(x)$ means there is exactly one element in the universe for which $\text{P}$ is true.
Which of the following statements is(are) Valid ?
- $\exists!x \text{P}(x) \wedge \exists!x \text{Q}(x) \Rightarrow \exists!x (\text{P}(x) \wedge \text{Q}(x))$
- $\exists!x (\text{P}(x) \wedge \text{Q}(x)) \Rightarrow \exists!x \text{P}(x) \wedge \exists!x \text{Q}(x)$
- $\exists!x \text{P}(x) \vee \exists!x \text{Q}(x) \Rightarrow \exists!x (\text{P}(x) \vee \text{Q}(x))$
- $\exists!x (\text{P}(x) \vee \text{Q}(x)) \Rightarrow \exists!x \text{P}(x) \vee \exists!x \text{Q}(x)$
- I, II, IV
- I, III
- II, III, IV
- IV only
