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From the table, we conclude a → b = a' ∨ b, as both columns have the same values (1, 1, 0, 1) for the same combinations of
a and b. This law is called Conditional elimination.
Example 3: Prove using truth table a ↔ b = (a → b) ∧ (b → a).
Ans.
a b a ↔ b a → b b → a (a → b) ∧ (b → a)
0 0 1 1 1 1
0 1 0 1 0 0
1 0 0 0 1 0
1 1 1 1 1 1
Both columns (a → b) ∧ (b → a) and a ↔ b have same values (1, 0, 0, 1) for the same combinations of a and b. Hence
proved. This rule is also called Biconditional elimination.
Example 4: Prove using truth table a → b = b' → a'.
Ans. a b a' b' a → b b' → a'
0 0 1 1 1 1
0 1 1 0 1 1
1 0 0 1 0 0
1 1 0 0 1 1
Columns a → b and b' → a' have identical values (1, 1, 0, 1), hence they are equivalent. This rule is also called
transportation (logic).
Example 5: Prove that (a ∧ (a → b)) → b = 1.
Ans.
a b a → b a ∧ (a → b) (a ∧ (a → b)) → b
0 0 1 0 1
0 1 1 0 1
1 0 0 0 1
1 1 1 1 1
Here, (a ∧ (a → b)) → b has value (1, 1, 1, 1) for all combinations of a and b. Hence, we conclude (a ∧ (a → b)) → b = 1.
The above rule is also called Modus Ponens.
3.6 TAUTOLOGY, CONTRADICTION AND CONTINGENCY
Tautology Contradiction Contingency
A compound proposition A compound proposition A compound proposition
is called a tautology if is called a contradiction if is called a contingency if
and only if it is true for all and only if it is false for all and only if it is neither a
possible truth values of its possible truth values of its tautology nor a contradiction.
propositional variables. The propositional variables. The The proposition contains both
proposition is evaluated as proposition evaluates as 0 0 (false) and 1 (true) in its
1 or true in the last column or false in its last column of last column of the truth
of the truth table. the truth table. table.
7676 Touchpad Computer Science-XI

