$f(x) = x$ or a , if $f(x)$ and $a$ is known find $x$ boolean algebra

Question

I am new to boolean algebra. I am facing difficulty solving this problem:

Given $f(x) = x \lor a$, for some $f(x)$ and $a$, deduce the value of $x$.

Can someone provide me the solution with example?

Similarly, if $p(x) = x \land a$ for some $p(x)$ and $a$, how can I deduce $x$?

Many thanks in advance.

Answer 1

If $Y=X \cup A$ then obviously $A \subseteq Y$.

For every subset $R \subseteq A$ the disjoint union $X:=R \cup (Y\setminus A)$ is a solution of $Y=X \cup A$, so there are $2^{\#A}$ solutions