# Working out if a predicate formula is true or false given a domain {0, 1}

I can't figure out how to interpret this. Is my understanding of the statement correct?

"There exists an x such that for all y, if p(x) is true then x = y"

i. P(0) = true, if y = 1 then x != y, formula is not true. p(1) = true, if y = 0 then, x != y, this formula is not true.

ii. P(0) = true, same as above. P(1) = false, since the premise is false the statement should be true. But of course x != y if y = 0

Same for iii.

Am I sort of making sense?

• You are sort of making sense. A little clearer writing and you can completely make sense. – астон вілла олоф мэллбэрг Jan 11 '17 at 9:36
• Your reading of the formula is correct. – Mauro ALLEGRANZA Jan 11 '17 at 9:41
• You have to consider in turn each sub-question... (i) both $P(0)$ and $P(1)$ are true. Thus, what happens to the formula "There exists an $x$ such that for all $y$, if $P(x)$ is true then $x = y$" ? – Mauro ALLEGRANZA Jan 11 '17 at 9:42
• There are only two possbile value for the "initial" $x$ : $0$ and $1$. Consider then in order: for $x=0$ is it true that "for all $y$, if $P(0)$ is true then $0=y$" ? – Mauro ALLEGRANZA Jan 11 '17 at 9:44
• Again, two possible choices for $y$ : $0$ and $1$. With $y=1$ we have $P(0) \to 0=1$; antecedent T and consequen F: thus the conditional is F. So it is not true for all $y$... – Mauro ALLEGRANZA Jan 11 '17 at 9:46