How to prove that () Δ () ≠ ( Δ ) for any 2 sets. So the question is how can I prove that for any 2 Sets, A and B, the symmetric difference between the power set of A and the power set of B IS NOT equal to the powr set of A symmetric difference B.
() Δ () ≠ ( Δ )
I have been stuck on this question for a while and I can't find a way to prove it.
The idea is that the empty set ($\varnothing$) lies in the RHS, but not in LHS. And thus the assertion is proved.
In details, let $A$ and $B$ be a set. Since any power set contains the empty set, we have that $\varnothing \in P(A),P(B),P(A\Delta B)$. By the definition of symmetric difference, we have $\varnothing \notin P(A)\Delta P(B)$. Hence, $P(A)\Delta P(B)\neq P(A\Delta B)$, as desired. QED
The symmetric difference of two sets is the set of elements that are in both sets but not in their intersection and is given by:
$$ A Δ B = A \cup B \backslash A \cap B $$
The powerset of a set is the set of all elements of $S$ including the empty set and the set itself and is represented by $P(S)$.
The cardinality of $P(S)$ is given by $$|P(S)| = 2^{|S|}$$
We need to prove that $$() \space Δ \space () ≠ ( \space Δ \space )$$
$$P(A \cup B) \backslash P(A \cap B) ≠  P((A \cup B) \backslash (A \cap B)) $$
Let $X = A \cup B$ and $ Y = A \cap B$. We need to show that:
$$P(X) \space \backslash \space P(Y) ≠ P(X \backslash Y)$$
We can prove that two sets are not equal by showing that they have a different number of elements. Let $|X| =x, |Y|=y$.
$$|X \backslash Y| = x - y$$
$$|P(X) \space \backslash \space P(Y)| = 2^x - 2^y$$
Similarly, on the right hand side,
$$|P(X \backslash Y)| = 2^{x-y}$$
We can prove that $$2^x - 2^y ≠ 2^{x-y}$$ by using a counter example. Suppose $x=1, y=2$
$$2^1 - 2^2 = -3 ≠ 2^{1-2}=1/2$$