Lecture 1 - Binary and Boolean Algebra Flashcards
Understand Binary and Boolean Algebra
Define the term “Analog”
Infinite number of increments and/or possibilities
Define the term “Digital”
Means that there is a limited set of values by which something can be represented
What does 0 represent in the binary number system?
ON
What does 1 represent in the binary number system?
OFF
MSB
Refers to the outermost left bit and stands for “Most Significant Bit”
LSB
Refers to the outermost right bit and stands for “Least Significant Bit”
How is the binary number system different from the decimal number system?
Where the decimal number system has 10 symbols to be represented with (0 to 9) the binary number system only has two (0 to 1)
Nibble
Group of 4 bits
Byte
Group of 8 bits
Word
Another group of bits where the length is processor dependant (depends on what processor we’re looking at)
How many numbers can be represented in a n-bit word?
The number of states which can be represented are: 2^n ; (0 - 2^n)
What formula is used to calculate the amount of bits needed to represent a number?
Since the highest number to be represented is found using the formula: y = 2^n - 1, therefore to find the number of bits to represent it: n = log[2] (y + 1) ; log base 2
What do the 0s and 1s represent in a digital signal?
They represent a range of voltages (since the signals are waves, as long as the voltage is within a range it will either be on or off)
Grey Zone
Interpretable region in a digital signal graph which is outside of the ON voltage and OFF voltage range (sandwiched in between)
Accuracy (Analog vs. Digital)
Analog: 0.1% is considered a very good accuracy
Digital: Accuracy is directly proportional to the amount of circuitry added
Long Term Storage (Analog vs. Digital)
Analog: Store data as voltage on a capacitor (short term for an approximate maximum of 30 mins)
Digital: Stores data in memory (long term can last up to years)
Speed (Analog vs. Digital)
Most common use of analog or digital
Analog: Fastest circuits, high-frequency, etc
Define “Analog Signals”
Smooth, continuous waves that can have any value within a range, like turning up the volume smoothly.
Define “Digital Signals”
Step-like signals that are either on or off, high or low, representing information using distinct levels, like flipping a light switch.
What is Boolean Algebra
Two-value algebra; values of either 0 or 1
AND
(ab) = X is 1 if both a and b are 1
OR
(a + b) = X is 1 if a or b or both are 1
XOR
X is 1 if exactly one of a or b are 1 (not both; essentially OR with an extra condition)
NAND
X is 0 if both a and b are 1
NOR
X is 0 if either a or b or both are 1
X-NOR
X is 0 if exactly one of a or b are 1
Absorption Laws (S1)
XY + X = X
Dual Absorption Laws (S2)
(X + Y) X = X
Basic Laws (B1) and (B2) and their Dual Form (B3) and (B4)
X + 0 = X (B1)
X + 1 = 1 (B2)
X (1) = X (B3)
X (0) = 0 (B4)
Idempotent (I1)
A variable is unchanged by operating with itself
X + X = X
Dual Idempotent (I2)
X (X) = X
Commutative Laws (C1)
X + Y = Y + X
Dual Commutative Laws (C2)
(X) (Y) = (Y) (X)
Associative Laws (A1)
X + (Y + Z) = (X + Y) + Z
Dual Associative Laws (A2)
(X) (YZ) = (XY) (Z)
Distributive Laws (D1)
(X + Y) (Z) = (XZ) + (YZ)
Dual Distributive Laws (D2)
(XY) + Z = (X + Z) (Y + Z)
