Logic Circuits

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Logic Circuits

• More on Logic Gates and Boolean Expressions

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
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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

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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

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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.

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

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

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

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Exercises

• We will work on
• Setting up Boolean Expressions from problem
  statements
• Reducing Boolean Expressions to determine output
• Stepping through given logic circuits to
  determine outputs for given inputs

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Boolean Expressions

• Example 1
• To get into a physics program in a university,
  Jane needs to have (University-Level) UL-Physics
  and either UL-Algebra or UL-Calculus. Assign
  Boolean variables to the conditions and write a
  Boolean expression for the physics program
  requirements.
• To get you started
• Let P represent whether Jane has UL-physics,
• Let A represent whether Jane has UL-algebra,
• Let C represent whether Jane has UL-calculus.

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Boolean Expressions Example 1
To get into a physics program in a university,
Jane needs to have (University-Level) UL-Physics
and either UL-Algebra or UL-Calculus. Assign
Boolean variables to the conditions and write a
Boolean expression for the physics program
requirements.

• P and (A or C)
• (P and A) or (P and C)

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Evaluate the Expressions

• A 1, B 0, C 1, D 0, E 1
• (A and B)
• (A or B)
• (A or D) and C
• (C and A and D) or E
• (C and A and D) and E
• Note C is not(C)

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Expressions Answers

• A 1, B 0, C 1, D 0, E 1
• (A and B) 1 AND 0 0
• (A or B) 1 OR 0 1
• (A or D) and C (1 OR 0) AND 1 1
• (C and A and D) or E (1 AND 1 AND 0) OR 1 1
• (C and A and D) and E (0 AND 1 AND 1) AND 1
  0

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Alarm System

• You are installing an alarm system in a house.
  The system has a panel with an on/off switch. If
  the system (panel switch) is off then the alarm
  will never sound. If the system is on, then the
  alarm will sound if the door sensor detects a
  problem or if the window sensor detects a
  problem.
• Can you write a Boolean expression for
  determining if the alarm is on (true) or off
  (false)?

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Alarm System

• Assume
• A 0panel switch off, 1panel switch on
• B 0no door problem, 1problem at door
• C 0no window problem, 1problem at window
• Logic
• IF A1 THEN
• IF B1 OR C1 THEN Z1 ELSE Z0
• Expression
• Z A and (B or C)

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Car Buzzer

• You are designing a system for allowing buzzer to
  be turned on in a car. If the car is turned off,
  then the buzzer should not function. If the car
  is turned on, and the door is open, or, the
  seat-belt is not fastened, the buzzer should be
  turned on or, if the car is turned on and if the
  trunk is not shut, or the fuel tank cap is not
  shut, the buzzer should be turned on.

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Car Buzzer

• A 0Car off, 1Car on
• B 0Door closed, 1 Door open
• C 0Seat-belt not fastened, 1Seat-belt fastened
• D 0Trunk closed, 1Trunk open
• E 0Fuel-tank cap closed, 1Fuel-tank cap open

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Car Buzzer

• One answer is
• Z (A and (B or C)) or (A and (D or E))

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Logic Circuits

• Question 1
• The given circuit has 3 logic gates AND, NOT,
  XOR. For what values of inputs A, B is the output
  1?

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Logic Circuits
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Logic Circuits

• Question 2
• The given circuit has 3 logic gates AND, OR,
  XOR. For what values of the inputs A, B, C is the
  output 1?

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Logic Circuits