Symmetric difference – equality proofs.

elementary-set-theorylogicproof-writing

I have a couple of statements to prove (self-learning, not homework). I'm not able to proceed with any of them (I've tried starting from RHS, LHS, etc.). I suppose I'm missing something. I would like to receive some tips or a solution for one of them and I will try to solve the rest.

The symmetric difference is defined as follows: $A \triangle B = (A \setminus B) \cup (B \setminus A)$.

$(A \cup B) \triangle C = (A \triangle C) \triangle (B \setminus A)$.

$(A \cap B) \triangle C = (A \triangle C) \triangle (A \setminus B)$.

$(A \setminus B) \triangle C = (A \triangle C) \triangle (A \cap B)$.

Best Answer

E.g. the first one; try to show two inclusions.

Suppose $x \in (A\cup B)\Delta C$. Then there are two cases:

  1. $x \in A \cup B$ and $x \notin C$. Then subcase a: $x \in A$ and then $x \in A \Delta C$ (as $x \in A \setminus C$) while $x \notin B\setminus A$ (as $x \in A$), and so $x \in (A \Delta C)\Delta (B \setminus A)$. Or subcase b: $x \in B$ while $x \notin A$ and then $x \in B \setminus A$ and $x \notin A \Delta C$ (as $x \notin C$ and $x \notin A$). Again $x \in (A \Delta C)\Delta (B \setminus A)$.
  2. $x \in C$ or $x \notin A \cup B$. This is easier, as then $x \in C$, $x \notin A$ and $x \notin B$ so $x \in A \Delta C$ (from $C \setminus A$) and $x \notin B \setminus A$ (from $x \notin B$) and again $x \in (A \Delta C)\Delta (B \setminus A)$.

So the left to right inclusion has been shown now.

The right to left inclusion has similar cases.

Try it, and then the rest. Verify the claims first by drawing a Venn diagram for three sets in general position.

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