i need help with logic please

[david]

1/ Let A be a non-empty set. Let P(A) denote the powerset of A. The relation R on
P(A) is given by BRC iff B ∩ C /= ∅ for B,C∈ P(A). Prove or disprove that R is
an equivalence relation on P(A). If R is an equivalence Relation find the equivalence class if A = {0, 1, 2}


2/ The relation T on R ( real numbers) is given by xTy iff x−y ∈ Q (rationa numbers). Prove that T is an equivalence relation and find the equivalence class of 0 and √3

 

[pka]

1/ Let A be a non-empty set. Let P(A) denote the powerset of A. The relation R on
P(A) is given by BRC iff B ∩ C /= ∅ for B,C∈ P(A). Prove or disprove that R is an equivalence relation on P(A). If R is an equivalence Relation find the equivalence class if A = {0, 1, 2}

If \(\displaystyle B\in\mathcal{P}(A)~\&~B\ne\emptyset\) then is it possible to have \(\displaystyle B\mathcal{R}B~?\)
SO?

2/ The relation T on R ( real numbers) is given by xTy iff x−y ∈ Q (rationa numbers). Prove that T is an equivalence relation and find the equivalence class of 0 and √3

It is easy to show that \(\displaystyle \mathcal{T}\) is reflexive, symmetric and transitive.
I will give you one of the equivalence classes:
\(\displaystyle 0/\mathcal{T}=\mathbb{Q}\), the rationals.