Why sum of minterms or product of maxterms gives us the boolean function?

In summary, the proposition x'y + y'x is a function that is true only in the cases where the rows of the truth table make it 1.
  • #1
Avichal
295
0
I have thought about it and every-time I think I have an answer I try to explain it to myself and I fail. I want an intuition behind it and if there is a proof better.

Thank you
 
Physics news on Phys.org
  • #2
You might get an answer if you state the question clearly.
 
  • #3
Stephen Tashi said:
You might get an answer if you state the question clearly.
Oh okay, sorry if it was not clear
For example say two Boolean expressions x and y: its function is given
x y F
0 0 0
0 1 1
1 0 1
1 1 0

To find the function we need to add the minterms which are x'y and y'x. So function is x'y+y'x.
On what is minterms the link is here

But I don't understand why adding the minterms gives us the function.
 
  • #4
I'll suggest an intuitive way.

Visualize a Venn diagram where we have draw overlapping circles X and Y on a piece of paper. Smaller areas on the paper can be described by "coordinates" that tell whether the area is in-or-out of each set. So the possible coordinates in the descriptions are:

[itex] X \cap Y [/itex]
[itex]]X \cap Y^c [/itex]
[itex] X^c \cap Y [/itex]
[itex] X^c \cap y^c [/itex]

Any area that you can make using only some the above pieces can be written as a union of some of the pieces.

Of course you could draw an irregular area on the page that could not be described by the above procedure. For example, you could draw an area that was partly in [itex] X \cup Y [/itex] and partly out of [itex] X \cup Y [/itex]. Such an area would not be "a function of" X and Y.

Returning to propositions, if a propositional function is a function of propositions x and y then it has a truth table. In the left columns of the table are listed all combinations of the truth and falsity values of x and y. The function is 1 precisely in the cases where the rows of its truth table make it 1. So writing the function as something like x'y + y'x amounts to saying the function is true on the rows of the truth table where entries of the leftmost two columns show the truth of x'y or y'x and it isn't true on any other rows.
 
  • #5
for your question. The concept of minterms and maxterms comes from the representation of boolean functions in terms of logical operations. Minterms and maxterms are two different ways of representing boolean functions, and they both have their own advantages and uses.

Minterms are the simplest form of boolean functions, where each term represents a unique combination of inputs that results in a specific output. For example, in a boolean function with two inputs, there are four possible combinations of inputs (00, 01, 10, 11) and each of these combinations can be represented as a minterm. By combining these minterms with logical operations (AND, OR, NOT), we can create a boolean function that accurately represents the desired output for each input combination.

On the other hand, maxterms are the complements of minterms. They represent the combinations of inputs that result in a specific output of 0, instead of 1. This means that by combining maxterms with logical operations, we can also create a boolean function that accurately represents the desired output for each input combination.

Now, why do we use the sum of minterms or product of maxterms to represent boolean functions? This is because these two representations allow us to easily manipulate and simplify boolean functions. By using the sum of minterms or product of maxterms, we can use boolean algebra and logical laws to simplify the function and make it more efficient.

For example, the sum of minterms representation allows us to use the distributive law to break down the function into smaller and simpler parts, making it easier to analyze and understand. The product of maxterms representation, on the other hand, allows us to use De Morgan's laws to simplify the function.

In summary, the use of sum of minterms or product of maxterms to represent boolean functions is based on the principle of simplification. These representations allow us to manipulate and simplify boolean functions, making them more efficient and easier to work with. I hope this explanation helps to provide some intuition behind the use of minterms and maxterms.
 

1. Why do we use the sum of minterms or product of maxterms to represent boolean functions?

The sum of minterms and product of maxterms are two common methods used to represent boolean functions because they provide a systematic and structured way to express complex boolean expressions. These methods also allow for easy conversion between different representations, such as between truth tables and boolean expressions.

2. How does the sum of minterms and product of maxterms relate to logic gates?

The sum of minterms and product of maxterms can be used to simplify boolean expressions and minimize the number of logic gates needed to implement a given circuit. This is because each term in the sum or product corresponds to a unique combination of inputs that can be represented by a single logic gate.

3. What is the difference between the sum of minterms and product of maxterms?

The sum of minterms is a boolean expression in which all possible minterms (i.e. rows) of a truth table that result in a 1 are ORed together. On the other hand, the product of maxterms is a boolean expression in which all possible maxterms (i.e. rows) that result in a 0 are ANDed together. In essence, the sum of minterms represents a boolean function in disjunctive normal form (DNF) while the product of maxterms represents a boolean function in conjunctive normal form (CNF).

4. How do we determine whether to use the sum of minterms or product of maxterms?

The decision between using the sum of minterms or product of maxterms often depends on the given boolean expression and the desired outcome. In general, the sum of minterms is used to emphasize the 1s (true outputs) in a boolean expression, while the product of maxterms is used to emphasize the 0s (false outputs). The choice also depends on the ease of conversion between different representations and the simplicity of the final expression.

5. Can the sum of minterms and product of maxterms be used interchangeably?

Yes, the sum of minterms and product of maxterms can be used interchangeably as they both represent the same boolean function. This is because any boolean function can be expressed in either DNF or CNF form, and therefore can be represented by either the sum of minterms or product of maxterms. The choice between the two is usually based on convenience and simplicity of representation.

Similar threads

Replies
1
Views
680
Replies
2
Views
590
  • Set Theory, Logic, Probability, Statistics
2
Replies
40
Views
6K
  • Set Theory, Logic, Probability, Statistics
Replies
10
Views
892
  • Engineering and Comp Sci Homework Help
Replies
1
Views
857
  • Quantum Interpretations and Foundations
Replies
13
Views
639
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
2K
  • Precalculus Mathematics Homework Help
Replies
3
Views
757
  • Linear and Abstract Algebra
Replies
2
Views
565
  • Calculus and Beyond Homework Help
Replies
2
Views
470
Back
Top