Question
(A → B) (~ A v ~ B)

Answer

Truth table:
Column $\rightarrow$ 1 2 3 4 5 6 7
Row $\downarrow$ A B ~A ~ B A$\rightarrow$B ~ A v ~ B (A$\rightarrow$B) $\rightarrow$ (~ A v ~ B)
1 T T F F T F F
2 T F F T F T T
3 F T T F T T T
4 F F T T T T T
      1(~) 2(~) 1, 2 ($\rightarrow$) 3, 4 (v) 5, 6 ($\rightarrow$)
Explanation: (A → B) → (~ A v ~ B) is the column no. 1 and 2 are the first pillars, while the remaining five are the secondary pillars. Column no. 7 presents the whole complex joint statement. Column no. 7 The following facts become clear:
(1) According to the first row, if A is true and B is true, then (A → B) → (~ A v ~ B) the whole statement is untrue.
(2) According to the second row, if A is true and B is false, then (A A B) → (~ A v ~ B) is the whole statement true.
(3) According to the third row, if A is untrue and B is true, then (A → B) → (~ A v ~ B) is the whole statement is true.
(4) According to the fourth row, if A is untrue and B is untrue, then (A → B) → (~ A v ~ B) the whole statement is true.

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