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I have been given that the following Boolean operation results in "True" and I have been asked to either provide the truth values of $A, B, C$ or explain why it cannot be deduced.

$$\left(A \rightarrow \left(B\lor C\right)\right)\land \left(\left(B\lor C\right)\rightarrow A\right)$$

My Attempt:

By conceiving the operation as $a \land b$ we get that $a$ and $b$ must correspond to truth value of $1$. So both $\left(A\rightarrow\left(B\lor C\right)\right)$ and $\left(\left(B\lor C\right)\rightarrow A\right)$ must be $1$. But now conceiving both operations as $c\rightarrow d$ we get three cases wherein the truth value is $1$. How to go about solving this? Is there a general strategy for solving this type of problems? Thanks

Paras Khosla
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1 Answers1

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You can rewrite $(A \rightarrow (B \lor C)) \wedge ((B \lor C) \rightarrow A) $ as $A \leftrightarrow (B \vee C)$ and this is true if and only if both $A$ and $B \lor C$ are true or both are false. Hence we can have:
i) $A,B,C$ are all false
ii) $A$ is true and at least one of $B$ and $C$ is true

user289143
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