obtain the truth table of xy+y'z+x'y'+x'z and then implement the function using logic gates
Question
obtain the truth table of xy+y'z+x'y'+x'z and then implement the function using logic gates
Solution
Sure, let's break this down step by step.
First, let's define the variables:
- x, y, and z are boolean variables.
- The ' symbol denotes the NOT operation.
- The + symbol denotes the OR operation.
- The absence of a symbol between variables denotes the AND operation.
The function is: f = xy + y'z + x'y' + x'z
Step 1: Truth Table
The truth table for the function is as follows:
| x | y | z | xy | y'z | x'y' | x'z | f |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
Step 2: Implement the function using logic gates
The function can be implemented using AND, OR, and NOT gates as follows:
- Connect x and y to an AND gate (G1), the output is xy.
- Connect y to a NOT gate (G2), and then connect the output of G2 and z to an AND gate (G3), the output is y'z.
- Connect x to a NOT gate (G4), and then connect the output of G4 and the output of G2 to an AND gate (G5), the output is x'y'.
- Connect the output of G4 and z to an AND gate (G6), the output is x'z.
- Finally, connect the outputs of G1, G3, G5, and G6 to an OR gate (G7), the output is f.
This is a simplified implementation and may be optimized further.
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