Boolean Algebra

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This material was developed with funding from the
National Science Foundation under Grant # DUE 1601612

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1000

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Binary Systems

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1110

0100

0

Binary Numbers

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0011

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0

Binary Input
Switches

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0000

Binary Output
LEDs

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15

1100

1001

1111

0111

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The position of the switches or state of the LEDs can be used to represent small and large numbers. Switches would be used to input binary data while LEDs could be used to output binary data. Let's convert binary numbers to decimal numbers. First identifying ones and zeros in each of the columns. The process requires you to place the binary numbers in the table. Starting from the right digit, each binary digital will occupy an adjacent column. If a 1 occupies the column we record the value if a zero occupies the column we do not record that value. This is how to count to 15 in Binary.

0010

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1010

1101

0101

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0001

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1011

1

Truth Values

True

In the study of mathematics, Boolean algebra is the branch of mathematics that focuses on mathematical and logical operations using binary numbers or values dealing with truth values, true and false, usually denoted 1 and 0. The ones and zeros can represent switch positions ON (1) or OFF (0) or LEDs ON (1) OFF (0). Binary math is based on the binary number system this is a number system that only consist of two digit 1 + 0 the use of these digits can be used to represent any number. Like the decimal system each digit in the binary system represents a value but these are values are based multiples or powers of 2 not multiples of 10. So the first digit on the left of the decimal point in the binary number system would represent 20 = 1. Each preceding digit to the left would represent a higher value or power of 2. For examples: 21=2, 22=4, 23=8, 24=16 and so on and so on.

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False

Binary Mathematics &
Boolean Algebra

25

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24

23

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128

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64

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32

27

16

Like the decimal system each digit in the binary system represents a value but these values are based multiples or powers of 2 not multiples of 10. So the first digit on the left of the decimal point in the binary number system would represent 20 = 1. Each number preceding digit to the left would represent a higher value or power of 2. For examples: 21=2, 22=4, 23=8, 24=16 and so on and so on.

26

8

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0.25

2-3

Going to the right is a little different. The first digit to the right of the decimal point would represent 2-1 = ½, then 2-2=1/4, 2-3=1/8, 2-4=1/16 and so on.

0.5

0.125

0.0625

Click switches on and off

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2-4

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2-1

0.00

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DP

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Boolean Expression:
X = A × B

INPUTS

AND

OUTPUT

A

X

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AND Gate

The AND gate implements logical conjunction. The AND gate must have at least two inputs, it can have many more, but only one output. The operations requires all input states to be HIGH (1) or true in order to produce a HIGH (1) output. The truth table represents the AND gate operation.

B

NAND

The NAND gate implements logical opposite of a conjunction. The NAND gate must have at least two inputs, it can have many more, but only one output. The operations requires all input states to be HIGH(1) or true in order to produce a LOW (0) output. The NAND gate is simple a AND gate with an NOT gate on the output. The truth table represents the AND gate operation.

Boolean Expression:
X = A × B

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NAND Gate

Boolean Expression:
X = A + B

The OR gate implements logical disjunction. The OR gate must have at least two inputs, it can have many more, but only one output. The operations requires only one input states to be HIGH (1) or true in order to produce a HIGH (1) output. The truth table represents the OR gate operation.

OR Gate

OR

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NOR Gate

Boolean Expression:
X = A + B

NOR

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The NOR gate implements the opposite of the logical disjunction. The OR gate must have at least two inputs, it can have many more, but only one output. The operations requires only one input states to be HIGH (1) or true in order to produce a LOW (0) output. An NOR is just an OR with an inverted output. The truth table represents the NOR gate operation.

XOR gate is a digital logic gate that gives a HIGH (1) true output when the number of HIGH inputs is odd. An XOR gate implements an exclusive OR; that is, a HIGH output results if one, and only one, of the inputs to the gate is HIGH. If both inputs are LOW(0) or both are HIGH, a LOW output results. A way to remember XOR is "one or the other but not both". The truth table represents the XOR gate operation.

XOR Gate

Boolean Expression:
X = A ⊕ B

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XOR

Boolean Expression:
X = A ⊕ B

XNOR Gate

XNOR gate is a digital logic gate that gives a LOW(0) false output when the number of HIGH inputs is odd. An XNOR gate implements an exclusive NOR; that is, a LOW output results if one, and only one, of the inputs to the gate is HIGH. If both inputs are LOW(0) or both are HIGH, a HIGH output results. An XNOR is just an XOR with an inverted output. The truth table represents the XNOR gate operation.

XNOR

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NOT Gate - (Inverter)

Boolean Expression:
A = X

Click switch on and off

The first logic gate is the NOT gate or Inverter. A NOT gate can only have one input and one output. An inverter circuit outputs a logic state representing the opposite logic-state to its input. Its main function is to invert the input state applied. The Boolean expression and operating truth table is shown below.

NOT

IEC Symbol

XNOR

Boolean algebra is based on logic operations using binary values. The laws of Boolean Algebra specifies seven logical operations. These operations are typically represented with symbols known as logic gates. Logic gates are the elementary building block of a digital circuit, smart devices and modern computers. Logic gates can have two or more inputs and one output.
There are two sets of symbols for elementary logic gates in common use, the "American Standards Association"(ANSI) which uses the distinctive shapes based on traditional schematics and the more modern square symbols used by the International Electrotechnical Commission (IEC).

Click to reveal each symbol.

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ANSI Symbol

Boolean Logic Gates