To develop a truth table for the given sum-of-products (SOP) expression X = AB + ABC + AC + ABC, we need to consider all possible combinations of inputs for variables A and B.
Let's create a truth table with columns for A, B, AB, ABC, AC, and X. Here's how we can fill it out step-by-step:
1. Start by listing all possible combinations of A and B. Since there are two variables, A and B, there will be four possible combinations:
A | B
0 | 0
0 | 1
1 | 0
1 | 1
2. Next, fill in the column for AB, which represents the logical AND operation between A and B. Multiply the values of A and B for each row to find the result:
A | B | AB
0 | 0 | 0
0 | 1 | 0
1 | 0 | 0
1 | 1 | 1
3. Fill in the column for ABC, which represents the logical AND operation between AB and C. Since C is not given in the expression, we assume C to be true (1):
A | B | AB | ABC
0 | 0 | 0 | 0
0 | 1 | 0 | 0
1 | 0 | 0 | 0
1 | 1 | 1 | 1
4. Fill in the column for AC, which represents the logical AND operation between A and C. Since C is assumed to be true (1), AC will be equal to A:
A | B | AB | ABC | AC
0 | 0 | 0 | 0 | 0
0 | 1 | 0 | 0 | 0
1 | 0 | 0 | 0 | 1
1 | 1 | 1 | 1 | 1
5. Finally, fill in the column for X, which represents the sum of the terms AB, ABC, AC, and ABC. Add the values of AB, ABC, AC, and ABC for each row to find the result:
A | B | AB | ABC | AC | X
0 | 0 | 0 | 0 | 0 | 0
0 | 1 | 0 | 0 | 0 | 0
1 | 0 | 0 | 0 | 1 | 1
1 | 1 | 1 | 1 | 1 | 3
The truth table for the SOP expression X = AB + ABC + AC + ABC is as follows:
A | B | AB | ABC | AC | X
0 | 0 | 0 | 0 | 0 | 0
0 | 1 | 0 | 0 | 0 | 0
1 | 0 | 0 | 0 | 1 | 1
1 | 1 | 1 | 1 | 1 | 3
In the truth table, the column for X shows the result of the SOP expression for each combination of inputs A and B.