Mister Exam

Expression ¬(¬((A↔B)∧(¬A→(¬B∨C))→C))

    The solution

    You have entered [src]
    ¬(¬(((a⇔b)∧((¬a)⇒(c∨(¬b))))⇒c))
    $$\neg \left(\left(\left(a ⇔ b\right) \wedge \left(\neg a \Rightarrow \left(c \vee \neg b\right)\right)\right) \not\Rightarrow c\right)$$
    Detail solution
    $$a ⇔ b = \left(a \wedge b\right) \vee \left(\neg a \wedge \neg b\right)$$
    $$\neg a \Rightarrow \left(c \vee \neg b\right) = a \vee c \vee \neg b$$
    $$\left(a ⇔ b\right) \wedge \left(\neg a \Rightarrow \left(c \vee \neg b\right)\right) = \left(a \wedge b\right) \vee \left(\neg a \wedge \neg b\right)$$
    $$\left(\left(a ⇔ b\right) \wedge \left(\neg a \Rightarrow \left(c \vee \neg b\right)\right)\right) \Rightarrow c = c \vee \left(a \wedge \neg b\right) \vee \left(b \wedge \neg a\right)$$
    $$\left(\left(a ⇔ b\right) \wedge \left(\neg a \Rightarrow \left(c \vee \neg b\right)\right)\right) \not\Rightarrow c = \neg c \wedge \left(a \vee \neg b\right) \wedge \left(b \vee \neg a\right)$$
    $$\neg \left(\left(\left(a ⇔ b\right) \wedge \left(\neg a \Rightarrow \left(c \vee \neg b\right)\right)\right) \not\Rightarrow c\right) = c \vee \left(a \wedge \neg b\right) \vee \left(b \wedge \neg a\right)$$
    Simplification [src]
    $$c \vee \left(a \wedge \neg b\right) \vee \left(b \wedge \neg a\right)$$
    c∨(a∧(¬b))∨(b∧(¬a))
    Truth table
    +---+---+---+--------+
    | a | b | c | result |
    +===+===+===+========+
    | 0 | 0 | 0 | 0      |
    +---+---+---+--------+
    | 0 | 0 | 1 | 1      |
    +---+---+---+--------+
    | 0 | 1 | 0 | 1      |
    +---+---+---+--------+
    | 0 | 1 | 1 | 1      |
    +---+---+---+--------+
    | 1 | 0 | 0 | 1      |
    +---+---+---+--------+
    | 1 | 0 | 1 | 1      |
    +---+---+---+--------+
    | 1 | 1 | 0 | 0      |
    +---+---+---+--------+
    | 1 | 1 | 1 | 1      |
    +---+---+---+--------+
    CNF [src]
    $$\left(a \vee b \vee c\right) \wedge \left(a \vee c \vee \neg a\right) \wedge \left(b \vee c \vee \neg b\right) \wedge \left(c \vee \neg a \vee \neg b\right)$$
    (a∨b∨c)∧(a∨c∨(¬a))∧(b∨c∨(¬b))∧(c∨(¬a)∨(¬b))
    DNF [src]
    Already transformed to DNF
    $$c \vee \left(a \wedge \neg b\right) \vee \left(b \wedge \neg a\right)$$
    c∨(a∧(¬b))∨(b∧(¬a))
    PCNF [src]
    $$\left(a \vee b \vee c\right) \wedge \left(c \vee \neg a \vee \neg b\right)$$
    (a∨b∨c)∧(c∨(¬a)∨(¬b))
    PDNF [src]
    $$c \vee \left(a \wedge \neg b\right) \vee \left(b \wedge \neg a\right)$$
    c∨(a∧(¬b))∨(b∧(¬a))
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