Question

Determine the SOP function that implements the truth table below and simplify it algebraically. Use the graphical method (using Bubble pushing and De Morgan's theorem) to change the function so that only NAND gate are required.

Bepaal die SOP funksie wat die onderstaande waarheidstabel implementeer en vereenvoudig dit algebvales. Gebruik die grafiese metode (met borre'stoting en De Morgan se stelling) om dhe funksie to verander sodat slegs NENhekke benodig word
x1 x2 x3 f
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1

a. f=(x₂​)(xˉ₁​​xˉ₃​)​
b. f=(x₁​)(xˉ₂​​xˉ₃​)​
c. f=(x₁)(xˉ₂​xˉ₃​)
d. f=(x₂​)(xˉ₁​xˉ₃​)
e. f=(x₂​)(x₁​​x3₃)​

Answer 1

The Sum of Products (SOP) function from the given truth table is simplified to f = (x2 ∧ ¬x3) + (x1 ∧ x3). By using De Morgan's theorem and bubble pushing, the function can be rewritten to only use NAND gates, resulting in f = ¬(¬(x2 ∧ ¬x3) ∧ ¬(x1 ∧ x3)).

Step-by-step explanation:

To determine the Sum of Products (SOP) function that implements the given truth table and simplify it algebraically, we first identify the minterms for which the function f is true (f=1). The product (AND) of literals is a minterm. Each minterm corresponds to a row in the truth table where the function output is 1. The SOP function is the sum (OR) of these minterms.

From the truth table, we can see that the function f is 1 for the following combinations of inputs x1 x2 x3:

• 0 1 0
• 0 1 1
• 1 0 1
• 1 1 1

Therefore, the SOP function is:
f = (¬x1 ∧ x2 ∧ ¬x3) + (¬x1 ∧ x2 ∧ x3) + (x1 ∧ ¬x2 ∧ x3) + (x1 ∧ x2 ∧ x3)

To simplify this function algebraically, we apply Boolean algebra rules. f can be simplified to:

f = (x2 ∧ ¬x3) + (x1 ∧ x3)

For the graphical method using only NAND gates, which are universal gates, we first express the function in a form that allows the direct use of NAND gates by pushing bubbles and applying De Morgan's theorem.

Express the simplified function as a NAND operation by applying De Morgan's theorem:

f = ¬(¬(x2 ∧ ¬x3) ∧ ¬(x1 ∧ x3))

This expression now only requires NAND gates for implementation.