- #1
nobahar
- 497
- 2
Hello!
1) [tex]\forall x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex]
2) [tex]\exists x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex]
3) [tex]\forall x (F(x) \land \forall y (F(y) \rightarrow y=x))[/tex]
4) [tex]\exists x (F(x) \land \forall y (F(y) \rightarrow y=x))[/tex]
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 [tex]\forall y (F(y) \rightarrow y=x))[/tex] 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 [tex]\forall y (F(y) \rightarrow y=x))[/tex] is false then:
3) and 4) are always false.
I understand [tex](F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex] to mean that if F(x) is true then [tex]\forall y (F(y) \rightarrow y=x))[/tex] 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 [tex](F(x) \land \forall y (F(y) \rightarrow y=x))[/tex] to mean that they are independent, and F(x) and [tex]\forall y (F(y) \rightarrow y=x))[/tex] can be true or false separately.
Is this correct?
Thanks in advance.
1) [tex]\forall x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex]
2) [tex]\exists x (F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex]
3) [tex]\forall x (F(x) \land \forall y (F(y) \rightarrow y=x))[/tex]
4) [tex]\exists x (F(x) \land \forall y (F(y) \rightarrow y=x))[/tex]
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 [tex]\forall y (F(y) \rightarrow y=x))[/tex] 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 [tex]\forall y (F(y) \rightarrow y=x))[/tex] is false then:
3) and 4) are always false.
I understand [tex](F(x) \rightarrow \forall y (F(y) \rightarrow y=x))[/tex] to mean that if F(x) is true then [tex]\forall y (F(y) \rightarrow y=x))[/tex] 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 [tex](F(x) \land \forall y (F(y) \rightarrow y=x))[/tex] to mean that they are independent, and F(x) and [tex]\forall y (F(y) \rightarrow y=x))[/tex] can be true or false separately.
Is this correct?
Thanks in advance.