How to determine whether the formulas A ∧ (B ⊕ C) and (A ∧ B) ⊕ (A ∧ C) are equivalent?

Question

How to determine whether the formulas A ∧ (B ⊕ C) and (A ∧ B) ⊕ (A ∧ C) are equivalent, without using a truth table?

Answer 1

The soundness theorem, ie. If $\Gamma\vdash\phi$, then $\Gamma\vDash\phi$. This means that if you can prove something (syntax side of things), then it is also true semantically (in terms of a truth table).

So if you can prove that $A ∧ (B ⊕ C)$ iff $ (A ∧ B) ⊕ (A ∧ C)$ using the proof system in the language (be it axiomatic proof/Gentzen style proof...etc), then the soundness theorem tells you that in terms of truth values the two formulas are equivalent.

PS: Welcome to Math SE! While I did not down vote your question, most questions are expected to show a certain level of effort of attempting (eg. show what you have worked out so far and where you got stuck), this way the community can better guide you through the question. Even if you are completely stuck with zero clue, telling us what is confusing you will help us to identify the next possible step.

Answer 2

Note: XOR is obviously a $\underline{\text{negation of equivalence}}$ so this is intuitive, also, there is analogy between XOR and $\underline{\text{symmetric difference}}$ of sets:

$$A\land(B\oplus C)\equiv A\land\neg(B\iff C)\equiv\neg\left(A\implies (B\iff C)\right)$$

$$(A\land B)\oplus(A\land C)\equiv\neg\left((A\land B)\iff(A\land C)\right)$$

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little hints:

$$A\oplus B\equiv(A\land\neg B)\lor(B\land\neg A)\;\;{\sim}\,\;A\Delta B\equiv\left( A\cap B^c\right)\cup\left(B\cap A^c\right)$$ $$(A\lor B)\land\neg(A\land B)\;\;{\sim}\;\; (A\cup B)\cap (A\cap B)^c$$