# Logic statements

1. Nov 5, 2011

### nobahar

Hello!

1) $$\forall x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))$$
2) $$\exists x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))$$
3) $$\forall x (F(x) \land \forall y (F(y) \rightarrow y=x))$$
4) $$\exists x (F(x) \land \forall y (F(y) \rightarrow y=x))$$

If 1) is true then 2) is true; if 1) is false then 2) may or my not be true
If 2) is true then 1) may or may not be true; if 2) is false then 1) is false

If $$\forall y (F(y) \rightarrow y=x))$$ is true then:
If 1) is true then 2) is true; if 1) is false then 2) may or my not be true
If 2) is true then 1) may or may not be true; if 2) is false then 1) is false

If $$\forall y (F(y) \rightarrow y=x))$$ is false then:
3) and 4) are always false.

I understand $$(F(x) \rightarrow \forall y (F(y) \rightarrow y=x))$$ to mean that if F(x) is true then $$\forall y (F(y) \rightarrow y=x))$$ is true. So the first two are determined by whether or not all x or there is some x that make F(x) true. I understand $$(F(x) \land \forall y (F(y) \rightarrow y=x))$$ to mean that they are independent, and F(x) and $$\forall y (F(y) \rightarrow y=x))$$ can be true or false separately.
Is this correct?