Question

Show that symmetric sentences are independent from reflexive, and transitive sentences by constructing a structure that satisfy transitive and reflexive quality but not symmetric.

Reflexive :∀xE(x, x)
symmetric :∀xy(E(x, y) → E(y, x))

transitive: ∀xyz(E(x, y) ∧ E(y, z) → E(x, z))

Answer 1

Symmetric sentences are independent from reflexive, and transitive sentences, let us consider an example

E(x,y) : x divides y and take domain as set of integers

Then it is reflexive because x divides x for all x.

It is transitive because let E(x, y) ∧ E(y, z) then there exists two integers k_{​​​​​​1} and k_{​​​​​​2} such that y=xk_{​​​​​1} and z =yk_{​​​​​2} , then

z=(xk_{​​​​​1})k_{​​​​​​2} implies x divides z i.e E(x,z).

But it is not symmetric because E(2,4) is true but E(4,2) is not true.

2 divides 4 but 4 doesn't divides 2.