 In this video, we're going to explain what is known as the canonical representation of a Boolean expression. And for that, we start with a Boolean expression, in this case x, y plus z prime, and from this Boolean expression, we derive its truth table, which is represented here. The truth table enumerates all possible 8 combinations when the symbols x, y and z take the possible values 0 and 1. And this column over here is the output or the result of the evaluation of this expression for all these possible combinations. So the first canonical representation we are going to propose is what is known as sum of products. The sum of product representation is obtained following the following three rules. The first one is we only consider rows with out equal to 1. So in this case, we'll all consider only the first, third, fifth, seventh and eighth rows. Then for each row, we're going to write a product, or in Boolean terms, a conjunction. But the product must follow this rule, which is if the value is 0 of the symbol, then the symbol appears negated. And if the value is 1, then the symbol appears without negation. So let's do it for this table over here. And let's start with this first row. This is the first row that contains a 1. All symbols appear 0, 0, 0. All of them are 0. So the rule says that if the value is 0, the symbol should appear in the product negated. Therefore, the product derived from this first row is x prime, y prime, z prime. We go to the third row, skip the second because it has a 0 here. In this case, however, y does not appear as a 0, appears as a 1. Therefore, its literal is not going to be negated. The resulting product is going to be x prime, y, z prime. Fifth row, something similar, although in this case it's x, the one that has the value 1. Therefore, the derived product is going to be x, y prime, z prime. We go to this row, the seventh row, x and y has value 1. Therefore, x and y don't appear negated, but z appears negated. And the final one, all of them are equal to 1. The product is going to be x, y, z. So again, we only consider rows with output equal 1. Each row will transform into a product with the rule that if the value is 0, then the symbol appears with a negation. If the value is 1, the symbol appears without a negation. And in all the products, all the symbols appear. The final step to obtain the sum of products is all products are in a sum. So the final touch here is we put an add here, add here, add here and add here. And this expression we have obtained is the sum of products or SOP of the Boolean expression we see here. Analogously, the second canonical representation is called the product of sums. And again, this representation is obtained following three similar steps, although not exactly the same. In this case, the rows we're going to select are those rows with the out column equal 0. So in this case, we're just going to select three rows from this truth table, which are the opposites that the one we selected before. The rule is also slightly different. For each row, we're going to write a sum or in Boolean logic terms, a disjunction. And the rule to apply when deciding how the symbols appear in this sum is that if the value is 1, then the symbol appears negative. And if value is 0, then the symbol appears without negation. So the rule is similar to the previous one, although with slight changes. So let's see what kind of sums we obtain like this. If we go to the second row, 001 here translates into X plus Y plus Z prime. If we go to the fourth row and we follow this rule is 011. If the value is 0, the symbol appears without negation. So in this case, it's going to be X plus Y prime plus Z prime. And this row over here, which is the third one without 0, will translate into the sum 101. Therefore, this is going to be X has the value 1, appears negated, X prime plus Y plus Z prime. And then the final step is that all sums are in a product, which is what we did by putting the parentheses around that. So the second notation here is what we call the product of sums. Now after doing this, we ended up with three expressions. Let's call this expression E1, let's call this expression E2, and this expression E3. Now these three expressions are very interesting because they are all related to this truth table. Remember, the Boolean expression made us derive this truth table. From this truth table, we derive the expression of the sum of products. And also from this truth table, we derive the expression of product of sums. So this is an example in which three expressions E1, E2, and E3, even though they are different, clearly different. This one and this one are clearly different. So are these two and these two as well. Even though they are different, they represent exactly the same Boolean function, the same truth table. Therefore, these three expressions, these three Boolean expressions, is what we call equivalent. However, when given a truth table, deriving a specific Boolean expression out of all possible ones, these two sum of products and products of sum is what they are known as canonical representations.