Consider the following in Boolean Algebra \(\begin{array}{l} \mathrm{X}: a…
2022
Consider the following in Boolean Algebra
\(\begin{array}{l} \mathrm{X}: a \vee(b \wedge(a \vee c))=(a \vee b) \wedge(a \vee c) \\ \mathrm{Y}: a \wedge(b \vee(a \wedge c))=(a \wedge b) \vee(a \wedge c) \\ a \vee(b \wedge c)=(a \vee b) \wedge c \text { is satisfied if } \\ \end{array}\)
- A.
X is true
- B.
Y is true
- C.
Both X and Y are true
- D.
It does not depend on X and Y
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Correct answer: C
Answer: Both X and Y are true. Below are concise derivations for each identity.
X: Start with a ∨ (b ∧ (a ∨ c)).
Apply distributivity inside: b ∧ (a ∨ c) = (a ∧ b) ∨ (b ∧ c).
Then a ∨ ((a ∧ b) ∨ (b ∧ c)) = a ∨ (b ∧ c), using absorption a ∨ (a ∧ b) = a.
Also (a ∨ b) ∧ (a ∨ c) = a ∨ (b ∧ c) by distributivity. Hence the two sides are equal for all a, b, c.
Y: Start with a ∧ (b ∨ (a ∧ c)).
Apply distributivity inside: b ∨ (a ∧ c) = (b ∨ a) ∧ (b ∨ c) = (a ∨ b) ∧ (b ∨ c).
Then a ∧ ((a ∨ b) ∧ (b ∨ c)) = a ∧ (b ∨ c), using absorption a ∧ (a ∨ b) = a.
Also (a ∧ b) ∨ (a ∧ c) = a ∧ (b ∨ c) by distributivity. Hence both sides are equal for all a, b, c.
Therefore both identities hold universally; the correct choice is that both X and Y are true.
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