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I'm asked to find a logical expression that is equivalent to the one listed in the question below, but I'm stumped as to what steps I would take next. If someone could show me step by step how to solve them and what rules would be used, I would really appreciate it.

  1. Using only the NOT and the AND operators, find an expression that is equivalent to ¬(a ∧ ¬b)↔𝑐

The furthest I got is (¬a∨b)↔c by using De Morgan's law, but I don't know how I can simplify it further than that. Someone told me the final answer is b ↔ c but I have no idea how they got there or if it's even correct?

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  • $\begingroup$ In part A) $b\leftrightarrow c$ is incorrect. What if $a$ and $b$ are flase and $c$ is true? $\endgroup$
    – saulspatz
    Jan 26, 2019 at 21:30
  • $\begingroup$ You want to remember that the biconditional can be broken up into two conditional statements with the AND: p $\leftrightarrow$ q $\equiv$ (p $\rightarrow$q) $\land$ (q $\rightarrow$ p). Then you use the implication equivalence and keep going. $\endgroup$
    – Joel Pereira
    Jan 26, 2019 at 21:39

1 Answer 1

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Since $A \leftrightarrow B$ is logically equivalent to $(\lnot A \lor B) \land (\lnot B \lor A)$, we have: \begin{align} \lnot (a \land \lnot b) \leftrightarrow c &\equiv ( \lnot \lnot (a \land \lnot b) \lor c) \land (\lnot c \lor \lnot (a \land \lnot b)) \\ &\equiv ((a \land \lnot b) \lor c) \land (\lnot c \lor \lnot (a \land \lnot b)) \\ &\equiv ((a \land \lnot b) \lor c) \land \lnot (c \land a \land \lnot b) \\ &\equiv ((a \lor c) \land (\lnot b \lor c)) \land \lnot (c \land a \land \lnot b) \\ &\equiv (\lnot (\lnot a \land \lnot c) \land \lnot (\lnot \lnot b \land \lnot c)) \land \lnot (c \land a \land \lnot b) \\ &\equiv (\lnot (\lnot a \land \lnot c) \land \lnot (b \land \lnot c)) \land \lnot (c \land a \land \lnot b) \\ \end{align}

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