first column is p
second column is q
third column is (p ^ q) which stands for p "and" q.
fourth column is (p v q) which stands for p "or" q.
fifth column is ~(p v q) which stands for "not" (p "or" q).
sixth column is (p ^ q) v (~(p v q)) which stands for:
(p and q) "or" (not (p and q).
The "or" logic is true for all conditions except when both variables involved in the "or" statement are false.
The "and" logic is false for all conditions except when both variables involved in the "and" statement are true.
The first row shows you the headings for each column as expressed in p and q.
The second row shows you the headings for each column as expressed in p, q, A, B, or C.
A is equal to (p ^ q).
B is equal to (p v q).
C is equal to ~(p v q).
The last column shows you (A v C) which translates to (p ^ q) v (~(p v q)).
The second row is not necessary, but i included it to show you that you can set another variable equal to a complex statement to make the statement more readable. If it confuses you, then just skip the second row and go with the headings in the first row as if the second row wasn't there.
The truth table has 4 rows to show all possible conditions for 2 variables.
The are 2 possible conditions for each variable involved. Since there are 2 variables involved, there are 2 * 2 = 4 possible conditions.
~(p v q) is the inverse of (p v q)
if a variable is true, then "not" that variable is false.
if a variable is false, then "not" that variable is true.
In the table, when (p v q) is true, then ~(p v q) is false, and vice versa.