Boolean Algebra 1 – The Laws of Boolean Algebra

This video is about the

laws

of Boolean

algebra

. You will see how these

laws

can be derived by examining simple logic circuits and considering some familiar properties of base 10 numbers. In the video that follows this one, you will see how the laws of Boolean

algebra

are applied. Boolean algebra can be applied to simplify complex Boolean expressions and you will have the opportunity to practice the skills involved before analyzing the laws of Boolean algebra in detail. Let's briefly consider why we need them. Here is a complex combination of logic gates and here is an expression. which describes the output Z in terms of the inputs to B and C.

This Boolean expression is an accurate description of this particular circuit. Boolean algebra can be applied step by step to reduce a complex expression like this to its simplest form. All of these expressions are equivalent, each one describes a different circuit, but all of these circuits do exactly the same thing. The behaviors of these circuits can be described with exactly the same truth table, so this combination of logic gates does exactly the same job as the complex combination, but it is clearly much simpler, it needs fewer components, so it is more reliable, it is faster, generates less heat, and is cheaper to manufacture because Boolean algebra can be used to simplify complex expressions and therefore the circuits they describe.

Boolean algebra is an extremely powerful tool for designing computers. Before you can use Boolean algebra to simplify a complex expression, you first need to know the laws, in fact, you must be so familiar with these laws that you can easily detect where and how can usefully apply, then you will need a lot of practice simplifying complex expressions to really master the skills involved, consider this or the door of one of the entrances is permanently set to one, there is always at least one entrance of one, so always There will be an output of one, regardless of the value of a with this and gate of one of the inputs is set to zero, so the output will always be zero, regardless of these expressions are collectively known as the nullification law.

You may have heard the term annulment in regards to a marriage that does not legally exist and that actually means to declare null and void, you can see in both cases that the entry a has no effect. Consider this or gate one of the inputs is always zero, so the output will always match the value of input a. If the input a is zero, we have two zeros, so a comes out zero if the input a is one, we have a 1 and a 0 coming in, so a 1 comes out and the gate layout is similar with one of the inputs permanently set to 1, the output will always match the input a, this pair of expressions is referred to collectively as the identity law, an input Ord with a zero or AND with a 1 will always give an output that is equal to the entrance.

Look at this layout here. Both inputs of gate o are always same if a is 1. inputs will be 1 so output will be 1, if input a is 0 then both inputs will be 0 so output will also be 0 of the same Thus, if both inputs of a gate and are always the same, then the output of the gate will match. This is known as the law of identity, simply put, an input that is odd with itself will result in an output that is equal to the input an input that is odd with itself will also result in an output that is equal to the input the following laws do not involve gates in this combination, the gate o is fed with the input a along with its inverse, so the gate o always receives a 1 and a 0, regardless of the value of a, the output of the gate o in this arrangement, therefore, must always be 1 in this combination, the y-gate is fed with the input a along with its inverse, so the y-gate always receives a 1 and a 0, regardless of what the output of the gate and here it must always be 0, another name for the inverse. of a Boolean value is its complement, so these expressions are known as the complement law.

Here we have two doors open in series if the input is a the output is no no a, if a is 1 the output is 1, if a is 0 the output is 0 in other words the output always matches the input , this is called the law of double and negation, in fact, any even number of non-serial gates will produce an output that matches the input. Now, at this stage, you may be thinking that Boolean expressions that describe combinations Logic gates typically do not include ones and zeros, they contain only letters and symbols that describe the outputs in terms of the inputs;

However, once you start applying the laws, you will see that some of the intermediate steps to simplify a complex expression may include ones and zeros that are eventually eliminated. Before continuing, let's take a look at the concepts of Boolean addition and Boolean multiplication. Consider this 0 plus 0 equals 0 of course, 0 plus 1 equals 1 one plus zero also equals one of course now in

boolean

logic. there is only true and false and the logic gate deals only with binary ones and zeros, so when it comes to one plus one it must equal something and that is certainly not zero, so one plus one equals one, in fact, logically speaking, one plus one plus one plus one plus one equals one look at the similarity between these sums and the truth table for a gate or we say that the

boolean

sum corresponds to the logical function of a gate or now look these multiplication operations, there are no surprises here at all compare it with the truth table for a gate and here you can see that boolean multiplication corresponds to the logical function of a gate and by the way, there is no such thing as boolean subtraction that would imply the existence of negative numbers, but there is only one and zero there is also no such thing as Boolean division because division is actually a repeated subtraction to derive the next set of laws.

Let's review some basic properties of base 10 numbers, namely associative, commutative and distributive properties. These properties are particularly relevant for students. of advanced mathematics, but you may have already applied them to base 10 arithmetic without a second thought. The associative property tells us something about the way numbers can be grouped. You know that 5 plus 2 plus 3 equals 10 and that 5 plus 2 plus 3 also equals 10, no matter which pair of values ​​you add first, you get the same result. This can be written algebraically as plus B plus C equals plus B plus C Now, because the edit corresponds to the Boolean operation u, we can say that a or B or C is equal to a or B or C the only difference between each side of the equation is the position of the brackets and they make no difference in terms of logic gates, we are saying that these two subtly different circuits behave in exactly the same way, similarly we know that 5 times 2 times 3 equals 30 and that 5 times 2 times 3 is also 30 algebraically we can write a times B times C equals a times B times C in base 10 algebra we tend to omit the multiplication symbols now because multiplication is like a boolean operation and it follows that a + b + C equals a and B and C to put it another way the behaviors of these two gate combinations are identical the commutative property refers to the way numbers can move in an expression, for example, 4 plus 8 equals a 12 and the same goes for 8 plus 4.

Algebraically we can say that a plus B equals B plus a the Boolean equivalent is a or B equals B or the logic gate equivalent of this equation is quite trivial, it really What we are saying is that there is no difference if we exchange the inputs of a door or in a similar example 4 times 8 is equal to 32 as is 8 times 4, therefore a times B is equal to B times a, so deduces that a and B are equal to B and a, again, we are really saying that it makes no difference if we swap the entries of an and the gate.

Moving on, you may have come across algebraic problems that require you to expand brackets or, conversely, factor problems out of an expression. Since these are based on the distributive property of numbers, for example, 2 times 3 plus 4 equals 2 times 3 plus 2 times 4 in regular algebra we can write a parenthesis B plus C equals b plus AC the Boolean equivalent is a and parentheses B or C is equal to brackets a and B or brackets a and C we can test the correctness of the so-called distributive law by comparing the circuits described on both sides of this equation, inspect this carefully and you will see that they behave in the same way that they share the same truth table, so the law is quite correct, we can also derive the related or distributive law in the same way this law has no equivalents in standard algebra, but this visual proof is convincing enough, our distributive law already is in force, now we can add another law, the so-called absorption law, also known as the redundancy law, the absorption law can also be proven with truth tables.

Note here that only when the input a is 1, the output can be 1, in fact the output is always the same as a, so if you can detect the term a and a or B within a complex expression, it can be simply replaced with a here to notice that the input a must be 1 along with 1 output, the output of this circuit is also always a, so the term a is in brackets a and B. it can be absorbed and replaced by these are the laws we have until now. The last law that we are going to mention is named after the esteemed friend and collaborator of George Bull or Gustus de Morgan.

The so-called Teorum of Demorgan. De Morgan used the Boolean rules. algebra that you have known now to prove that the complement of the product of two variables is equal to the sum of their complements in other words, he proved that this term is equivalent to this term when we examine the corresponding logic gate circuits and their truth. In the tables we can see that both sides of the equation are equal. De Morgan also showed that the complement of the sum of two variables is equal to the product of their complements. In other words, examination of this circuit equation and its truth tables reveals that this equation is also correct, so we now have a complete set of laws in the video that follows this one.

I'll show you how to start using them, make sure you have them on hand.