Given a binary relation R,S on set A, assume that R is anti-symmetric. Show R intersection S is anti-symmetric.
I started this proof by stating the definition of anti-symmetric with R which is
$$ ∀a∀b(((a, b) ∈ R ∧ (b, a) ∈ R) → (a = b))$$ I know the basic idea that since R does not have any elements that make it not anti-symmetric, (i.e. for any $(a,b) ∈ R$ (b,a) is not an element of R), any set $R ∩ S$ will not include elements that would make R not anti-symmetric because elements of the intersections have to include both sets. Because those same elements that would make R not antisymmetric any are not included in R, they would not be in $R ∩ S$. Even if S was not antisymmetric and it had elements (a,b) and (b, a) where a does not equal b, and (a,b) is common with R, $R ∩ S$ would not have have (b,a) thus making it antisymmetric. I know these things but I do not know how to write them into a formal proof. I am stuck at this.
Best Answer
It's best if you write down the property that $R\cap S$ must satisfy. You want to prove that $R\cap S$ is antisymmetric, so you want to show that
$$\forall a,b:((a,b)\in R\cap S \land (b,a)\in R\cap S\implies (a=b))$$
Now, you prove this claim like any other claim starting with $\forall$: