TER 3MI
– Comm Tech
Boolean Algebra Part 1
Mathematical rules are based on the limits we place on the particular numerical values we are dealing with. When we say that 1 + 1 = 2 or 3 + 4 = 7 , we are implying the use of integer quantities: the same types of numbers we all learned to count in elementary school. What most people assume to be self-evident rules of arithmetic -- valid at all times and for all purposes -- actually depend on what we define a number to be.
The Greek philosopher Aristotle founded a system of logic based on only two types of statements: true and false . His definition of truth led to the four laws of logic:
1. the Law of Identity (A is A);
2. the Law of Non-contradiction (A is not non-A);
3. the Law of the Excluded Middle (either A or non-A);
4. the Law of Rational Inference.
These so-called Laws function within the scope of logic where a statement is limited to one of two possible values , but may not apply in cases where statements can hold values other than "true" or "false."
The English mathematician George Boole developed several rules of relationship between mathematical quantities limited to one of two possible values: true or false, 1 or
0. His mathematical system became known as Boolean algebra. All arithmetic operations performed with Boolean quantities have only one of two possible outcomes: either 1 or 0. There is no such thing as "2" or "-1" or "1/2" in the Boolean world.
Let us begin our exploration of Boolean algebra by adding numbers together:
The first three sums make perfect sense to anyone familiar with elementary addition.
The last sum, though, is quite possibly responsible for more confusion than any other single statement in digital electronics, because it seems to run contrary to the basic principles of mathematics. Well, it does contradict principles of addition for real numbers, but not for Boolean numbers. Remember that in the world of Boolean algebra, there are only two possible values for any quantity and for any arithmetic operation: 1 or
0. There is no such thing as "2" within the scope of Boolean values.
Since the sum "1 + 1" certainly isn't 0, it must be 1 by process of elimination .

In class, we learned that there are some logical equivalents to the values true and false :
1 & 0, HI and LO, ON and OFF, OPEN and CLOSED
In our world of electronics, we are probably most concerned about ON and OFF because these correlate to the two positions of a switch, the two states of a transistor or the state of conduction/non-conduction.
You know that the following circuit will result in the light bulb being ON:
The Boolean expression illustrating the function of this circuit is 1 + 1 = 1. In other words, IF (SW1 is ON) AND (SW2 is ON) THEN (Lightbulb is ON).
An alternative could be:
The Boolean expression would be the same: 1 + 1 = 1, however the Logical Expression looks a bit different: IF (SW1 is ON) OR (SW2 is ON) THEN (Lightbulb is ON).
Now, imagine that one of the switches is opened or turned ‘OFF’:
The Boolean expression becomes 0 + 1 = 1, because if either switch is on, then the bulb will be lit! The alternative expression, 1 + 0 = 1 would be used if SW1 was closed and SW2 was open.
Finally, if BOTH SW1 and SW2 are open (or OFF), then the Boolean expression would be 0 + 0 = 0.