Mister Exam
-
How to use it?
- Expression:
- (P⊕Q)∧((P→R)∨(Q→S))
- ¬(A∧B)∨¬(¬A∨¬B)∧¬(A∨B)
- (a∧b)∨(b∧c)
- (p↔~q)⇒(~p∧q)∨(p∧~q)
-
Identical expressions
- (P⊕Q)∧((P→R)∨(Q→S))
- P⊕Q∧P→R∨Q→S
Expression (P⊕Q)∧((P→R)∨(Q→S))
⌨
The solution
You have entered
[src]
(p⊕q)∧((p⇒r)∨(q⇒s))
$$\left(\left(p \Rightarrow r\right) \vee \left(q \Rightarrow s\right)\right) \wedge \left(p ⊕ q\right)$$
Detail solution
$$p ⊕ q = \left(p \wedge \neg q\right) \vee \left(q \wedge \neg p\right)$$
$$p \Rightarrow r = r \vee \neg p$$
$$q \Rightarrow s = s \vee \neg q$$
$$\left(p \Rightarrow r\right) \vee \left(q \Rightarrow s\right) = r \vee s \vee \neg p \vee \neg q$$
$$\left(\left(p \Rightarrow r\right) \vee \left(q \Rightarrow s\right)\right) \wedge \left(p ⊕ q\right) = \left(p \wedge \neg q\right) \vee \left(q \wedge \neg p\right)$$
$$p \Rightarrow r = r \vee \neg p$$
$$q \Rightarrow s = s \vee \neg q$$
$$\left(p \Rightarrow r\right) \vee \left(q \Rightarrow s\right) = r \vee s \vee \neg p \vee \neg q$$
$$\left(\left(p \Rightarrow r\right) \vee \left(q \Rightarrow s\right)\right) \wedge \left(p ⊕ q\right) = \left(p \wedge \neg q\right) \vee \left(q \wedge \neg p\right)$$
Simplification
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$$\left(p \wedge \neg q\right) \vee \left(q \wedge \neg p\right)$$
(p∧(¬q))∨(q∧(¬p))
Truth table
+---+---+---+---+--------+ | p | q | r | s | result | +===+===+===+===+========+ | 0 | 0 | 0 | 0 | 0 | +---+---+---+---+--------+ | 0 | 0 | 0 | 1 | 0 | +---+---+---+---+--------+ | 0 | 0 | 1 | 0 | 0 | +---+---+---+---+--------+ | 0 | 0 | 1 | 1 | 0 | +---+---+---+---+--------+ | 0 | 1 | 0 | 0 | 1 | +---+---+---+---+--------+ | 0 | 1 | 0 | 1 | 1 | +---+---+---+---+--------+ | 0 | 1 | 1 | 0 | 1 | +---+---+---+---+--------+ | 0 | 1 | 1 | 1 | 1 | +---+---+---+---+--------+ | 1 | 0 | 0 | 0 | 1 | +---+---+---+---+--------+ | 1 | 0 | 0 | 1 | 1 | +---+---+---+---+--------+ | 1 | 0 | 1 | 0 | 1 | +---+---+---+---+--------+ | 1 | 0 | 1 | 1 | 1 | +---+---+---+---+--------+ | 1 | 1 | 0 | 0 | 0 | +---+---+---+---+--------+ | 1 | 1 | 0 | 1 | 0 | +---+---+---+---+--------+ | 1 | 1 | 1 | 0 | 0 | +---+---+---+---+--------+ | 1 | 1 | 1 | 1 | 0 | +---+---+---+---+--------+
CNF
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$$\left(p \vee q\right) \wedge \left(p \vee \neg p\right) \wedge \left(q \vee \neg q\right) \wedge \left(\neg p \vee \neg q\right)$$
(p∨q)∧(p∨(¬p))∧(q∨(¬q))∧((¬p)∨(¬q))
DNF
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Already transformed to DNF
$$\left(p \wedge \neg q\right) \vee \left(q \wedge \neg p\right)$$
(p∧(¬q))∨(q∧(¬p))