Set theory problem about sets that contain at most one element

Question

It is problem 9 in A.Shen' book

Consider an equality whose left-hand side and right- hand side contain set variables and operations ∩, ∪, \. Prove that if this equality is false for some sets, then it is false for some sets that contain at most one element.

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Answer 1

That's interesting - seems like it sys something about $\Bbb Z_2$-valued homomorphisms of Boolean algebras, sort of. Anyway, I bet the following works. First show this:

Lemma Fix $x$, and define $f(A)=A\cap\{x\}$ for all $A$. Then $f(A\cup B)=f(A)\cup f(B)$, $f(A\cap B)=f(A)\cap f(B)$ and $f(A\setminus B)=f(A)\setminus f(B)$ for all $A,B$.

Now say the two expressions are $\phi(A_1,\dots,A_n)$ and $\psi(A_1,\dots,A_n)$. Suppose $\phi(A_1,\dots,A_n)\ne\psi(A_1,\dots,A_n)$. Then there exists $x$ with $x\in\phi(A_1,\dots,A_n)$ and $x\notin\psi(A_1,\dots,,A_n)$ (or vice versa). Hence $$\phi((f(A_j))=f(\phi((A_j)))\ne f(\psi((A_j)))=\psi((f(A_j)).$$