obtain the truth table of xy+y'z+x'y'+x'z and then implement the function using logic gates

1. Break Down the Problem

We need to:

1. Obtain the truth table for the Boolean expression $f = xy + y'z + x'y' + x'z$.
2. Implement this function using logic gates.

2. Relevant Concepts

• Truth Table: A table that lists all possible values of variables (inputs) and the output of the function.
• Logic Gates: Basic building blocks of digital circuits, including AND, OR, and NOT gates.

3. Analysis and Detail

A. Truth Table Construction

1. Identify variables: $x, y, z$.
2. Determine the number of possible combinations (for 3 variables, this is $2^3 = 8$).
3. Calculate the output $f$ for each combination of $x, y, z$.

The final column $f$ shows the output for the function for each combination of inputs.

4. Verify and Summarize

• The truth table is complete.
• The function is true for inputs: (0,0,0), (0,0,1), (1,0,0), (1,0,1), (1,1,0), and (1,1,1).

Final Answer

The truth table for $f = xy + y'z + x'y' + x'z$ is fully constructed above.

Logic Gates Implementation

To implement $f$ using logic gates, the expression can be broken down as follows:

1. AND gates to produce $xy$, $y'z$, $x'y'$, and $x'z$.
2. OR gate to combine the outputs of the AND gates.

The circuit consists of:

• $AND$ gate for $xy$
• $NOT$ gates for $y$ and $z$ to get $y'$ and $z'$
• $AND$ gate for $y'z$
• $AND$ gate for $x'y'$
• $AND$ gate for $x'z$
• A final $OR$ gate combining the outputs from the four $AND$ gates to get $f$.