# Section 2.4: Problem 2 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
To which axiom groups, if any, do each of the following formulas belong?
(a) $[(\forall xPx\rightarrow\forall yPy)\rightarrow Pz]\rightarrow[\forall xPx\rightarrow(\forall yPy\rightarrow Pz)]$ .
(b) $\forall y[\forall x(Px\rightarrow Px)\rightarrow(Pc\rightarrow Pc)]$ .
(c) $\forall x\exists yPxy\rightarrow\exists yPyy$ .
(a) The formula is valid and has the form of $[(A\rightarrow B)\rightarrow C]\rightarrow[A\rightarrow(B\rightarrow C)]$ which is a tautology, group 1.
(b) The formula is a generalization of $\forall x(Px\rightarrow Px)\rightarrow(Pc\rightarrow Pc)$ and has the form of $\forall y[\forall x\alpha\rightarrow\alpha_{t}^{x}]$ where $t=c$ is substitutable for $x$ in $Px\rightarrow Px$ , group 2.
(c) This is not a valid formula, consider, for example, $\mathfrak{A}=(\mathbb{N};<)$ .