Truth tables are useful in discussing equivalent forms of statements. For example we can prove that p> q is equivalent to ~ p v q through the use of truth tables. We must first construct a truth table for each case.
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The column under p>q is equivalent to the column under ~Pv q proving that the two statements are equivalent. Statements can be substituted to further show equivalence. Let p be “I see” and q be “I learn.” The statement “If I see then I learn” is equivalent to “I do not see or I learn.” Referring back to the Pretest we can now prove that question two is true.
Try constructing truth tables for the following:
a. ~ p v q
b. p v ~ p
c. ~ (p q)
d. ~ (p v q)
Solutions:
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Prove that the following two statements are equivalent by constructing a truth table.
“It is not true that I race or that I do not win”=”I do not race and I win.”
~ (P v ~ q)= ~ p ^ q
~ q)= ~ p ^ q
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Since the tables for ~ (P v ~ q) and ~ p ^ q are identical we can say that the statements are equivalent. Also since the conditional statement (last row of table) is true in all cases we call it a tautology. A tautology is a statement which is always true. Statements can be tautologies without being equivalent if they are always true but not identical. This too is beyond the scope of this unit.
