Data Storage

1
Data Storage

• Binary Digits
• (The Story of Boole 1850s and Shannon 1938)

2
Electronic Speak

• In Paris they simply stared when I spoke to them
  in French I never did succeed in making those
  idiots understand their own language.
• Mark Twain, The Innocents Abroad, 1869
• Electronic Machine understands on and off, so,
  the Machine Alphabet is just two letters.

3
Boolean Logic in Everyday Life

• Situations
• You have a buzzer in your car that sounds when
  your keys are in the ignition and the door is
  open.
• Your fire alarm installed in your house will
  sound if it senses heat or smoke.
• There is a federal election coming up. People are
  allowed to vote if they are a U.S. citizen and
  they are 18.
• Mr. Boole is giving his students an assignment.
  To complete the assignment the students must do
  a presentation or write an essay.
• Stephanie is applying to a university Applied
  Mathematics program. She must have already taken
  Algebra, Calculus and Physics or Chemistry

4
Deduction One form of Logic

• Premise John is a human
• Premise Humans are greedy
• Conclusion John is greedy
• Deductions of this type is formally known as
  syllogistic logic, and the above assertion is
  known as a syllogism

5
Logic is

• concerned with the formal techniques of making
  conclusions from premises
• concerned with the validity of the argument,
  not the accuracy of the premises

6
Logic Question

• Humans have wings and can fly
• Jane is a human
• Therefore, Jane has wings and can fly
• Is this argument logical?
• Is this argument true?

7
Ambiguity in Logic

• Logic is focused on arguments, not expression.
  Logic does not address ambiguity of expression
• Mom You may have ice cream if you eat your
  dinner and do your homework or clean your room.
• May I have ice cream if I do not eat my dinner
  and I finish the homework?

8
ExerciseAmbiguity in Logic

• Question Given 1) or 2) below, would you be able
  to solve the problem What is the value of (2x -
  4) / (30 - 2y) ?
• 1) x 3
• 2) y gt 4
• Select an answer from the five choices a) - e).
• a) One (1) alone is sufficient to answer the
  question.
• b) Two (2) alone is sufficient to answer the
  question.
• c) Both alone are sufficient to answer the
  question.
• d) Together, both are sufficient to answer the
  question.
• e) Neither are sufficient to answer the question.

9
Algebra

• Informally, algebra is the branch of mathematics
  that works with expressions and variables
• More formally, an algebra is a set of axioms,
  operations, and values, i.e.
• A a,o,v

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Algebra

• You have already encountered normal algebra
• Operations Addition, Subtraction,
  Multiplication, Division
• Values 0, 1, 2, 3, 4, 5, 6, 7, 8,
• Axioms 1n n for all n
• 0n n for all n
• etc

11
Boolean Algebra

• Operations
• AND ( or something looking like multiplication)
• OR ( or something looking like addition)
• NOT ()
• Values
• TRUE (1)
• FALSE (0)

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Why study Boolean Algebra?

• Boolean algebra supports the notions of addition,
  subtraction, multiplication, and division
  familiar to you from normal arithmetic
• These arithmetic operations can be expressed in
  terms of Boolean AND, OR, and NOT operators using
  the binary representation of numbers

13
Notational Remarks

• Boolean Operators are often spelled out for
  clarity
• X Y X OR Y
• X Y X AND Y
• X NOT(X)

14
Truth Tables

• Truth tables are intended as an easy way to
  evaluate Boolean algebra operations and
  expressions
• Truth tables list the premises and conclusions.
  Premises are often referred to as input, and
  conclusions as output.
• Typical Truth Tables only deal with TRUE
  FALSE inputs which maybe represented by 1 or 0
  respectively.

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Truth Tables

• AND Operator

16
Truth Tables

• OR Operator

17
Truth Tables

• NOT Operator

18
Operands and Results

• Notice that AND and OR accepts two operands as
  input, and produces a single result
• The NOT operator accepts one operand as input,
  and produces a single result
• Operands Input Premises
• Results Output Conclusions

19
Expressions

• Algebraic expressions are a way to communicate a
  series of operations and the order in which they
  are performed
• Expressions are algorithms for producing a
  specific output from a specific input

20
Expressions

• Expressions use variables instead of actual
  values variables are considered placeholders
  (or, containers) that will assume specific
  values (or, contain the value) when the
  expression is evaluated
• Z (A B)

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Expressions

• The action of associating a specific value to a
  variable is known as binding.
• Expressions are evaluated by binding the
  variables and then performing the operations
  indicated

22
Expressions

• Parenthesis are NOT operators, but are included
  in expressions to indicate the order in which
  operations should be performed
• Expressions are evaluated inside out, i.e.
  operations inside parenthesis are performed
  first, and the results are used in subsequent
  steps of the evaluation

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Boolean Expressions

• Z (A or B)
• Z equals A or B
• A 1, B 0 ? Z 1
• Z (A and B) or A
• A 1, B 0 ? Z 1
• Z A and (B or A)
• A 1, B 0 ? Z ?

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Exercise Evaluating Expressions

• Purpose Practice evaluating expressions
• Who Pairs only
• Task Evaluate the Boolean expressions on the
  following slide. Find the value of the
  expression.
• Product Present value of expression when
  requested
• Time Limit 5 minutes

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Exercise Expressions

• A 1, B 0, C 1
• (A or B)
• (A or B) and C
• A and (B or C)
• ((not C) and A) or B

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Expressions Answer

• A 1, B 0, C 1
• (AB) 1 OR 0 1
• (AB)C (1 OR 0) AND 1 1
• A (BC) 1 OR (0 AND 1) 1
• (C A) B (0 AND 1) OR 0 0

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Algebra and Logic

• Algebra may be thought of as logic plus
• Boolean algebra can address some issues of
  ambiguity that exist in syllogistic logic

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Algebra and Logic

• Boolean algebra forces one to interpret the
  ambiguity one of the two candidate
  representations must be selected
• Both statements are unambiguous, but have
  different results

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Boolean Operations
Figure 1.1The Boolean operations AND, OR, and XOR
30
Computer Hardware

• At a hardware level a computer is a series of
  interconnected switches that either allow current
  to pass or prevent current from passing
• Switches are called transistors when you talk to
  a hardware engineer

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Transistors

• A transistor is an on/off switch with no moving
  parts except electrons, the movement of which
  constitutes an electric current

T
32
Controlling Transistors

• The switch state of a transistor changes when a
  current is applied to the control
• Transistors that are normally open will close
  when a current is applied
• Transistors that are normally closed will open
  when a current is applied

33
Gates

• Transistors are used to build components called
  logic gates, or simply gates
• Types of gates are
• AND gate
• OR gate
• NOT gate

34
Logic Gates AND, OR
Figure 1.2 (A)A pictorial representation of AND,
OR, XOR, and NOT gates as well as their input and
output values

35
Logic Gates XOR , NOT
Figure 1.2 (B)A pictorial representation of AND,
OR, XOR, and NOT gates as well as their input and
output values

36
Transistors and Boolean Logic

• In a normally open transistor, current flows if
  and only if both the In and Control
  components have current
• Out 1 if and only if In AND Control 1
• A normally open transistor implements the AND
  operator

37
Transistors and Boolean Logic

• In a normally closed transistor, current flows if
  and only if there is current applied to In and
  no current applied to Control. If In is set
  with a permanent current then Control is always
  the reverse of Out
• Control NOT Out or Out NOT Control
• A normally closed transistor implements a NOT
  operator

38
Transistors and Boolean Logic

• OR operator is not fundamental
• P OR Q is the same as
• NOT((NOT P) AND (NOT Q))
• OR operator can be implemented using a
  combination of AND and NOT operators

39
Practical Logic

• Contrariwise, continued Tweedledee, if it was
  so, it might be and if it were so, it would be
  but as it isnt, it aint. Thats logic.
• -Lewis Carroll, Alices Adventures in
  Wonderland, 1865

40
Gates and Expressions

• Logic gates are physical constructions that
  implement AND, OR, and NOT logical operations
• Logic gates are used to build electrical circuits
  that implement logical expressions
• Anything that can be represented by a logical
  expression can be implemented in a hardware
  circuit

41
Arithmetic Circuits

• Arithmetic operations can be represented as
  Boolean expressions, so hardware circuits can be
  built to add, subtract, multiply, and divide two
  numbers
• Numbers are represented in binary form, which we
  will discuss further in the next lecture

42
Circuit Simulator

• There are a few circuit simulators available from
  other universities online to explore some of
  these concepts. One is located at
• http//www.math-cs.gordon.edu/courses/cs111/module
  7/logic-sim/example1.html

43
Memory Circuits

• So, whats so special about all this?
• Boolean logic allows for the creation of
  circuits that remember their previous state.
  These are called flip flops and are discussed
  in the text.
• We will study the flip-flop example in the text,
  Fig 1.3., Page 22.

44
Flip-Flop Circuit
Figure 1.3A simple flip-flop circuit

45
Manipulating output from flip-flop
Figure 1.4 (A)Setting the output of a flip-flop
to 1

46
Setting the output of a flip-flop
Figure 1.4 (B)Setting the output of a flip-flop
to 1
47
Setting the output of flip-flop to 1
Figure 1.4 (C)Setting the output of a flip-flop
to 1