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Boolean Logic and Circuits. Professor James T. Williams, Jr. HONP112 Week 2 Lesson. George Boole (1815-1864). George Boole developed the mathematics that made modern computing possible, almost a century before the first computers were actually developed. . Boolean Logic. - PowerPoint PPT Presentation
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Boolean Logic and Circuits
Professor James T. Williams, Jr.HONP112
Week 2 Lesson
George Boole (1815-1864) George Boole
developed the mathematics that made modern computing possible, almost a century before the first computers were actually developed.
A special kind of mathematics where there are only 2 possible values.
These values are true and false. Special operators are used with these
values to create expressions. A boolean expression will evaluate to either
true or false in its entirety.
Boolean Logic
There are three Boolean operators: AND OR NOT
Boolean Operators
First, any expression in parenthesis Then the AND Then the OR
Order of Operations
The AND operator is a BINARY operator (i.e. it acts on two variables or expressions – “binary” in this context does not mean “base-2”)
Example expression: A AND B Also can be written as A * B The resulting value of the expression
depends on the values of A and B.
The AND Operator
An AND expression only evaluates to true if BOTH of the sub-expressions also evaluate to true. Otherwise, it evaluates to false.
false AND false = false false AND true = false true AND false = false true AND true = true
Evaluating AND expressions
AND – Truth Table We can look at all
the possibilities for the value of A*B as a truth table:
A B A*B
false false false
false true false
true false false
true true true
Assume: A is true and B is false What does the expression A * B evaluate to? What if A is true and B is true?
AND - Example
Assume: A is true, B is false, C is true, D is true
What does the expression A * B evaluate to? What does the expression C * D evaluate
to? What does (A*B) * (C*D) evaluate to?
AND – More examples
The OR operator is a BINARY operator (i.e. it acts on two variables or expressions)
Example expression: A OR B Also can be written as A + B (Careful – the
plus sign does not mean “and”!!) The resulting value of the expression
depends on the values of A and B.
The OR Operator
An OR expression only evaluates to true if EITHER of the sub-expressions also evaluate to true. Otherwise, it evaluates to false.
false OR false = false false OR true = true true OR false = true true OR true = true
Evaluating OR expressions
OR – Truth Table We can look at all
the possibilities for the value of A+B as a truth table:
A B A+B
false false false
false true true
true false true
true true true
Assume: A is true and B is false What does the expression A + B evaluate
to? What if A is true and B is true? What if A is false and B is false?
OR - Example
Assume: A is true, B is false, C is true, D is true
What does the expression A + B evaluate to?
What does the expression C + D evaluate to?
What does (A+B) + (C+D) evaluate to?
OR – More examples
The NOT operator is a UNARY operator (i.e. it acts on one variable or expressions)
Example expression: NOT A Also can be written as -A The resulting value of the expression
depends on the value of A
The NOT Operator
An NOT expression simply evaluates to the inverse value of the expression.
NOT false = true NOT true = false
Evaluating NOT expressions
NOT – Truth Table We can look at all
the possibilities for the value of -A as a truth table:
A -A
false true
true false
Assume: A is true and B is false What does the expression -A evaluate to? What does the expression -B evaluate to?
NOT - Example
Consider (B AND NOT A) OR (NOT D OR C). Assume A=1, B=1, C=1, and D=0 Put in the variable values and simplify as
below: (1 AND NOT 1) OR (NOT 0 OR 1) (1 AND 0) OR (1 OR 1) 0 OR 1 1
Evaluating Boolean Expressions
Some more examples to work out in class or on your own time. Remember the order of operations.
Assume A=1, B=1, C=1, and D=0 A OR D AND B = x (NOT B AND C) AND (A OR C) = x (B AND NOT D OR C) OR (NOT B AND B) = x (A AND B AND C) OR (B AND D) = x (A*B*C) + (B*D) = x [alt. version of above]
More Boolean Expressions
All computer/digital circuits are constructed from various combinations of Boolean expressions.
These are implemented in the computer by very tiny electronic components, built into chips, called “gates.”
Depending on the values going into the gates, the end result will be either a logical 0 or 1. This corresponds with false or true.
What does this have to do with computers?
The gates in a computer understand the state 0 or 1 based on electrical voltage.
Very generally speaking, for our purposes only, a 1 is about 5 volts, and a 0 is 0 volts or close to it.
In real life these values may vary, but the idea is the same.
It is more accurate to refer to the 1 and 0 in computer circuits as “high” or “low”.
Technically speaking…
In case you wonder where the following screen shots are coming from...
There is a free logic circuit simulator called Logisim.
http://ozark.hendrix.edu/~burch/logisim/ Dark green is low, bright green is high Let’s see what happens with our logic
gates.
The Logisim simulator
The AND gate is represented using the symbol below.
It takes two inputs and produces a single output. For the output to be high, both inputs must be high.
The AND gate
Let’s change one of the inputs to high. Because both are not high, the output is
still low.
The AND gate
Now let’s make both inputs high. Notice that now the output is high.
The AND gate
The OR gate is represented using the symbol below.
It takes two inputs and produces a single output. For the output to be high, only one of the inputs must be high.
The OR gate
Let’s change one of the inputs to high. Notice that the output went high just by
making one of the inputs high.
The OR gate
The symbol below represents the NOT gate. This is also called an “inverter.”
Notice that the output is high when the input is low.
The NOT gate
Changing the input to high makes the output low.
The NOT gate
(A AND NOT B) OR (C OR D) = x What is x? It depends on the values
assigned to A, B, C, and D. Remember the values can only be true or
false.
Consider This Boolean Expression
(A AND NOT B) OR (C OR D) = x Assume A=true, B=false, C = false, D=true.
Our Example as a Circuit
(A AND NOT B) OR (C OR D) = x Assume A=false, B=false, C = false,
D=true.
Let’s make a change
(A AND NOT B) OR (C OR D) = x Assume A=false, B=true, C = false,
D=false.
Let’s make another change
Maybe in class, or on your own time … try to visualize them as circuits this time around.
(A AND NOT B) AND (C OR D) (A * -B) * (C + D) [alt. version of above] (D + -B + C) * C -A * B * (C + A) Substitute different values for the variables. Try some with more or less variables.
Experiment. Learn by doing.
Some more examples …
In circuit design, there are some special gates that are commonly used.
The purpose of these gates is to make circuits simpler to build (less hardware = less effort = less cost).
The gates we will discuss are not standard boolean operators, but are actually single circuits constructed from the standard boolean gates.
Special Gates
NAND: “Not And” (AND gate followed by a NOT gate)
NOR: “Not Or” (OR gate followed by a NOT gate)
XOR: “Exclusive OR” (means that you only get a high output if EITHER of the inputs is high, not both. The circuit for XOR is more complex than for NAND or NOR)
Three Special Gates
Same as AND followed by a NOT. Notice that we only draw the bubble part of the NOT gate on a NAND symbol (short-cut)
The NAND gate
Same as OR followed by a NOT. Notice that we only draw the bubble part of the NOT gate on a NOR symbol (short-cut)
The NOR gate
This means that only one of the two inputs can be high to get a high output. Notice how XOR symbols are drawn, and see the two simulations below. (The XOR circuit itself is not shown).
The XOR gate
NAND – Truth Table Remember our
earlier AND truth table. Just invert the output values and that is the NAND truth table.
Let’s use 1 and 0 to represent true/high and false/low.
A B A NAND B
0 0 1
0 1 1
1 0 1
1 1 0
NOR – Truth Table Remember our
earlier OR truth table. Just invert the output values and that is the NOR truth table.
Let’s use 1 and 0 to represent true/high and false/low.
A B A NOR B
0 0 1
0 1 0
1 0 0
1 1 0
XOR – Truth Table Remember our
earlier OR truth table. XOR is the same except that we get low when both inputs are high.
Let’s use 1 and 0 to represent true/high and false/low.
A B A XOR B
0 0 0
0 1 1
1 0 1
1 1 0
Know the truth tables for the six gates we have discussed.
Know the various ways to represent the logical values of true/false.
Be able to evaluate a boolean expression using the three simple operators.
Be able to evaluate a circuit using any of the six logic gates.
Let’s stop and review
At this point, trying to imagine what the computer can do with these types of circuits may be a bit abstract.
So let’s look at a concrete example of a real circuit that is used by every computer.
You will not have to memorize this circuit but hopefully it will help illustrate a real-life application of a boolean logic circuit.
Boolean circuits in action…
This circuit (a “full adder”) is used by computers to add one column of two whole numbers (in base-2 of course).
Again … You do not have to memorize this circuit, but just try to understand what it does.
This is a real circuit
Imagine you are adding a single column of numbers. Notice there are three inputs. These are the first
addend, the second addend, and the current value of the carry.
There are two outputs. One is the result value that gets placed in the result column, and the other is the new carry value.
We already know the addition algorithm … so let’s test the circuit to make sure it works correctly.
Important: in the following slides, the “+” symbol will mean “plus” !
The Full Adder - analysis
Assume the carry is 0, the first addend is 1, the second addend is 0.
0+1+0 = 1. 1 is not >= the base, so we set the result to 1, and the carry stays zero.
Full adder – one test
Now, assume the carry is 0, the first addend is 1, the second addend is also 1.
0+1+1= 2. 2 is >= the base, so we subtract the base from the result, and set the carry to 1.
Full adder – another test
OK, now assume that the carry, the first addend, and the second addend are all 1.
1+1+1 = 3. 3 is >= the base, so we subtract the base from the result, and set the carry to 1.
Full adder – another test
A Question: By now you should be asking yourself “what if the two numbers we are adding are more than one column wide”? How does the adder handle multiple columns??
The answer: A whole bunch of individual full adders are “chained” together to account for many columns. When you move to the next column, the next full adder circuit takes in the two new addends, and also the carry value calculated from the previous column.
The Full Adder – a question
The adder is just one example of a logic circuit that is really used inside a computer. There are numerous and varied logic circuits that make up an entire computer.
Though we will not be examining the specific circuits, just keep the concept in mind as we continue our studies.
I hope this lecture has provided insight into how boolean logic is related to computers.
Our next lesson will expand on the boolean logic idea and how it applies to sets.
Conceptually speaking …