Exercises 14-23 deal with the Boolean algebra {0,1} with addition, multiplication, and complemen

Exercises 14-23 deal with the Boolean algebra {0,1} with...

Key Concepts

Boolean Algebra

Boolean algebra is a branch of algebra that deals with binary values, typically 0 and 1, and is used to model logical operations. It defines operations such as addition (logical OR), multiplication (logical AND), and complement (logical NOT), with each operation following specific axioms and properties that facilitate the analysis and design of digital circuits and logical expressions.

Associative Laws

The associative laws state that the grouping of operations does not affect the outcome. In the context of Boolean algebra, these laws ensure that for any elements a, b, and c, the expressions a + (b + c) and (a + b) + c yield the same result for addition, and a • (b • c) and (a • b) • c yield the same result for multiplication. Verifying these laws confirms the consistency and reliability of the algebraic operations.

Truth Tables

Truth tables are a systematic way of listing all possible combinations of input values for variables in Boolean expressions and the corresponding outputs. By organizing these values in a table format, one can easily verify properties such as the associative laws, as every possible scenario is covered, and any discrepancies would be immediately apparent. This method is particularly useful when dealing with the binary nature of Boolean algebra.

Transcript

00:01 All right, so we need to prove x plus y plus z is equal to x plus y plus z, and also need to prove that x, y z is equal to xy z.

00:29 So for problems like these, we're going to use the truth table, and the number of rows in the truth table is determined by two to the power of the number of variables.

00:40 In this case, we have three variables, x, y, z.

00:43 Meaning that this is going to be eight rows.

00:46 So let's do that.

00:48 We got our x, y, and z.

00:52 And we need to go through every scenario.

00:54 So every case where x is 0, every case is where x is 1, and same with y and z.

01:03 So we do 0 -0 -0 -0 -0 -1, 0 -1, 1 -0 -0 -1, 1 -0 -0 -0 -1, 1 -0 -0 -0 -0 -1, 110, and 111.

01:27 And since we're starting off with the additive law, we're going to need to check the y plus z as well as x plus y.

01:35 And then finally combine those with the respective third values so that we can form the expression that we're looking for.

01:46 So we're going to do y plus z, and then we're going to do x plus y plus z.

01:55 And then we're going to do x plus y and finally x plus y plus z so when we're looking at y plus z that means either y or z or both have to equal one in order for y plus z to be equal to one so we put zeros where both y and z are zero so here and here and then fill the rest with ones like so.

02:42 And while we're here, we can do the same thing with x and y.

02:45 Since x and y are both zero in the first two rows, we do zero and zero.

02:52 And then for all the other rows, since at least one of x and y are equal to one, we put ones for all of them, like so.

03:07 And then we're going to compare the fourth column, the y plus z, with the x.

03:14 And if at least one of them is a one, the result is going to be a 1.

03:19 So everything on the second, third, fourth, six, seventh, and eighth are all going to be 1.

03:27 And since x is 1 in the 5th row, that's also going to be 1.

03:30 Since both x and y plus z are 0 on the first row, that's going to be a 0.

03:38 And we're going to do the same thing with x plus y and also with z.

03:42 As long as at least one of them is 1, they're all going to be 1.

03:47 So 3rd, 4, 5th, 6, 6 ,000.

03:49 7th, then 8th are all ones because of the x plus y.