Question 7.8: Using a Truth Table to Prove a Boolean Expression Prove the ...
Using a Truth Table to Prove a Boolean Expression
Prove the associative law for the OR operation (Equation 7.8), which states that
(A + B) + C = A + (B + C)
The truth table listing all possible combinations of the variables and the values of both sides of Equation 7.8 is shown in Table 7.2. We can see from the truth table that A+(B+C) and (A+B)+C take the same logic values for all combinations of A, B, and C. Because both expressions yield the same results, the parentheses are not necessary, and we can write
A + (B + C) = (A + B) + C = A + B + C
Table 7.2. Truth Table Used to Prove the Associative Law for the OR Operation (Equation 7.8)
(A + B) + C = A + (B + C) = A + B + C (7.8)
| A | B | C | (A+B) | (B+C) | A+(B+C) | (A+B)+C | A+B+C |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
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