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I'm looking for some free resources online (books, courses, text, sites and so on) on how to solve these two particullar problems and understand the material behind them. There isn't any material on this in my native language and it seems quite hard to find any info when the I don't know the exact terminology in english.

Any help with these problems will be helpful :)

1)

Prove that the set of made of the following formulas is satisfiable ( realizable / I don't know the exact terminology in english)

$ \forall x \neg p(x,x)$

$ \forall x \forall y ( p(x,y) \Rightarrow \neg p(y,x))$

$\exists x \forall y (x \neq y \Rightarrow p(x,y))$

$\exists x \exists y ( x \neq y \Rightarrow p(x,y))$

2)

The structure $A$ has a bearer $N$ (the set of natural numbers) and it is for a language with only one non-logical symbol $p$, which is a predicate interpreted like : $ \in p^A \leftrightarrow n^2 = km + 1$ . Prove that $\{0\}$, $\{ 1 \}$ and $\{ : n \in N \}$ are definable.

Agian any help on finding material on how to solve these and understand them is of great help. Thanks :)

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    Online, you can see Stephen Simpson, [Mathematical Logic](http://www.personal.psu.edu/t20/notes/logic.pdf) (2013). – 2017-01-02
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    *Satisfiable* is the correct English term. – 2017-01-02
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    Note that your formulas include two copies of $$\exists x \forall y((x\neq y \rightarrow p(x,y))$$ Did you intend to write $\forall x\exists y((x\neq y)\rightarrow p(x, y))$? – 2017-01-02
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    @Mauro It just occurred to me that the OP has asked two separate questions. – 2017-01-02
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    In order to show that the set of formulas is *satisfiable* you choose a suitable interpretation and check that in it all the formuals are true. For the first one, you can consider e.g. $\mathbb N$ and $<$ for the predicate symbol $p$ and check that $∀n∈ \mathbb N \lnot (n < n)$ is true. – 2017-01-02
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    It seems to me that the suggested interpretation will work for all the four formulas of problem 1) ... – 2017-01-02
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    Regarding problem 2), the corerct term is *definable*: in order to prove e.g. that $\{ 0 \}$ is definable in the structure $A$, you have to show that you can build with $p$ a formula $\varphi$ such that $\varphi(n)$ is true in $A$ **iff** $n=0$. – 2017-01-02
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    You can try with $\varphi(m) := \exists n \ \forall k \ (n^2=km+1)$... – 2017-01-02
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    I second Mauro's recommendation, which is one of the free online references I listed in the linked post. – 2017-01-03

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