Find quotient set cardinilty of $|\{0,1\}^A/R|$

Question

Let A be finite set (not empty) and $B$ be subset of $A$

$fRg \iff ∀ x \in B : f(x) =g(x)$

($R$ Equivalence relation)

Any idea how to find $|\{0,1\}^A/R|$ when $B=\{\}$ and when $|B|=k$?

Answer 1

hint prove that the following function is a bijection: $\varphi : \{0,1\}^A / R \to \{0,1\}^B $ such that $$ \varphi([f]) = f_{|B} $$

1. $\varphi$ is well defined. If $f \sim g$, then $\forall x \in B \quad f(x) = g(x)$ which means that $f_{|B} = g_{|B}$ and so $\varphi([f]) = \varphi([g])$

2. $\varphi$ is iniective. $\varphi([f]) = \varphi([g]) \Rightarrow f_{|B} = g_{|B} \Rightarrow f \sim g \Rightarrow [f] = [g]$

3. $\varphi$ is suriective. Let $h: B \to \{0,1\}$ and define $$ f(x) = \begin{cases} 0 & \textrm{if} \: x \in A \setminus B \\ h(x) & \textrm{if} \: x \in B \end{cases} $$ Then $\varphi([f]) = f_{|B} = h$. So $\varphi$ is a bijection