Find the canonical sum of products and product of sums expression for the function
F = X1X2X3 + X1X3X4 + X1X2X4.
The correct answer and explanation is :
Let’s find the Canonical Sum of Products (SOP) and the Canonical Product of Sums (POS) for the function:
[
F = X_1 X_2 X_3 + X_1 X_3 X_4 + X_1 X_2 X_4.
]
1. Canonical Sum of Products (SOP)
In canonical SOP form, each product term (AND operation) must represent a minterm, which corresponds to a unique combination of the variables (X_1, X_2, X_3, X_4). To express (F) as a canonical SOP, we need to write all the minterms for which (F = 1).
Start by identifying the minterms for each term in the sum (X_1 X_2 X_3 + X_1 X_3 X_4 + X_1 X_2 X_4).
- For (X_1 X_2 X_3): This term is true when (X_1 = 1), (X_2 = 1), and (X_3 = 1), but we need to consider all combinations of (X_4). So the corresponding minterms are:
- (X_1 X_2 X_3 \cdot X_4)
- (X_1 X_2 X_3 \cdot \overline{X_4})
- For (X_1 X_3 X_4): This term is true when (X_1 = 1), (X_3 = 1), and (X_4 = 1), but we also need to consider (X_2). So the corresponding minterms are:
- (X_1 X_2 \cdot X_3 X_4)
- (X_1 \overline{X_2} \cdot X_3 X_4)
- For (X_1 X_2 X_4): This term is true when (X_1 = 1), (X_2 = 1), and (X_4 = 1), but we need to consider (X_3). So the corresponding minterms are:
- (X_1 X_2 \cdot X_3 X_4)
- (X_1 X_2 \cdot \overline{X_3} X_4)
Thus, the Canonical SOP is:
[
F = X_1 X_2 X_3 \cdot X_4 + X_1 X_2 X_3 \cdot \overline{X_4} + X_1 X_2 \cdot \overline{X_3} X_4 + X_1 \overline{X_2} \cdot X_3 X_4.
]
2. Canonical Product of Sums (POS)
In canonical POS form, each sum term (OR operation) must represent a maxterm, which corresponds to a unique combination of the variables (X_1, X_2, X_3, X_4). We need to write all the maxterms for which (F = 0).
To find the maxterms, first write the combinations where (F = 0). From the original expression, you can observe where (F = 0) by identifying the combinations of inputs that make (F) false. After deriving those combinations, we can form the maxterms, which are ORed terms involving negations of the variables.
After identifying the zeros in the truth table, we derive the maxterms and write the canonical POS form.
The Canonical POS form is:
[
F = (X_1 + X_2 + X_3 + \overline{X_4})(X_1 + X_2 + \overline{X_3} + X_4)(X_1 + \overline{X_2} + X_3 + \overline{X_4})(\overline{X_1} + X_2 + X_3 + X_4).
]
Summary:
- The Canonical SOP is the sum of all the minterms where the function evaluates to 1.
- The Canonical POS is the product of all the maxterms where the function evaluates to 0.
These canonical forms are essential in Boolean algebra, especially when designing circuits or simplifying logic expressions. They provide a systematic way to represent any Boolean function using only ANDs, ORs, and NOTs.