Show that no one of the following sentences is logically implied by the other two.

Question

The following question is from Herbert.B.Enderton, A Mathematical Introduction to Logic.

Section 2.2;Problem 2

Show that no one of the following sentences is logically implied by the other two. (This is done by giving a structure in which the sentence in question is false, while the other two are true.)

1.)$\forall x\ \forall y\ \forall z (Pxy \rightarrow (Pyz \rightarrow Pxz)$ This is transitivity

2.)$\forall x\ \forall y\ (Pxy\rightarrow (Pyx\rightarrow x= y))$ This is anti-symmetric

3.)$\forall x\ \exists y\ Pxy \rightarrow \exists y\ \forall x \ Pxy$

Minimally, my language requires ($P, \forall, =$). But here is where I'm stuck, how do I go about constructing a structure to answer the question ?

Answer 1

Consider the following relations on $\{a,b,c\}$.

• $\{(a,a),(a,b),(b,a),(b,b),(c,c)\}$
• $\{(a,a),(a,c),(b,a),(c,b)\}$
• $\{(a,b),(b,b),(b,c),(c,b)\}$

They respectively satisfy only 1, 2 and 3.