Mister Exam

# Expression (p⇒(q⇒r))⇒((p⇒q)⇒(q⇒r))

### The solution

You have entered [src]
(p⇒(q⇒r))⇒((p⇒q)⇒(q⇒r))
$$\left(p \Rightarrow \left(q \Rightarrow r\right)\right) \Rightarrow \left(\left(p \Rightarrow q\right) \Rightarrow \left(q \Rightarrow r\right)\right)$$
Detail solution
$$q \Rightarrow r = r \vee \neg q$$
$$p \Rightarrow \left(q \Rightarrow r\right) = r \vee \neg p \vee \neg q$$
$$p \Rightarrow q = q \vee \neg p$$
$$\left(p \Rightarrow q\right) \Rightarrow \left(q \Rightarrow r\right) = r \vee \neg q$$
$$\left(p \Rightarrow \left(q \Rightarrow r\right)\right) \Rightarrow \left(\left(p \Rightarrow q\right) \Rightarrow \left(q \Rightarrow r\right)\right) = p \vee r \vee \neg q$$
Simplification [src]
$$p \vee r \vee \neg q$$
p∨r∨(¬q)
Truth table
+---+---+---+--------+
| p | q | r | result |
+===+===+===+========+
| 0 | 0 | 0 | 1      |
+---+---+---+--------+
| 0 | 0 | 1 | 1      |
+---+---+---+--------+
| 0 | 1 | 0 | 0      |
+---+---+---+--------+
| 0 | 1 | 1 | 1      |
+---+---+---+--------+
| 1 | 0 | 0 | 1      |
+---+---+---+--------+
| 1 | 0 | 1 | 1      |
+---+---+---+--------+
| 1 | 1 | 0 | 1      |
+---+---+---+--------+
| 1 | 1 | 1 | 1      |
+---+---+---+--------+
DNF [src]
$$p \vee r \vee \neg q$$
p∨r∨(¬q)
PDNF [src]
$$p \vee r \vee \neg q$$
p∨r∨(¬q)
CNF [src]
$$p \vee r \vee \neg q$$
p∨r∨(¬q)
$$p \vee r \vee \neg q$$
p∨r∨(¬q)