Comp 3303 Elements of Computer Science
Boolean Algebra – Part 1
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Activities
A1 / Truth Tables and Mini-terms (F12)Here we shall work through a number of examples, for each we shall create the truth table, work out the mini-terms and create a boolean expression for the truth table.
In each example you have two switches A, B and an alarm light L. The switch when pressed gives a boolean output “1” else “0”. If the light gets a boolean “1” then it is on, else it is off.
Consider this problem. The light is on in either of two cases: When A is not pressed and when B is not pressed. Or when A is not pressed and B is pressed.
(a) Create the truth table for the system
A / B / L
0 / 0
0 / 1
1 / 0
1 / 1
(b) Now add in the mini-terms in the appropriate rows
A / B / L / mini-terms
0 / 0
0 / 1
1 / 0
1 / 1
(b) Now write down the boolean expression for L
(c) Use the logic simulator to build the circuit corresponding to the Boolean expression and check that its behaviour agrees with your initial truth table.
A2 / Truth Tables and Mini-terms (F3)
Repeat the above stages for this problem. The light is on in either of two cases: When A is pressed and when B is not pressed. Or when A is pressed and B is pressed.
A3 / Truth Tables and Mini-terms (F14)
Repeat the above stages for this problem. The light is on in either of three cases: When A is not pressed and when B is not pressed. Or when A is not pressed and B is pressed. Or when A is pressed and B is not pressed.
A4 / Truth Tables and Mini-terms (F13)
Repeat the above stages for this problem. The light is on in either of three cases: When A is not pressed and when B is not pressed. Or when A is not pressed and B is pressed. Or when A is pressed and B is pressed.
A5 / Truth Tables and Mini-terms (5)
Some safety critical systems such as aircraft flight controllers use three computers which process identical information, however it is understood that computers are not infallible and may fail. To mitigate against this a voting system is often used where the output is used only if two or more computers agree.
Let’s agree that the computers produce a logical output 0 or 1 and that logical 1 means a “correct” output.
(a) Build a truth table for the voting system
A / B / C
0 / 0 / 0
0 / 0 / 1
0 / 1 / 0
0 / 1 / 1
1 / 0 / 0
1 / 0 / 1
1 / 1 / 0
1 / 1 / 1
(b) Construct mini-terms for each line of the table where the output is 1. You should find 4 such mini-terms.
(c) Combine the mini-terms to produce a Boolean expression for the voting system
A6 / Three Input Problems
Repeat the above for the following system. The light is on in three cases: When A is pressed and B is not pressed, or when A is not pressed and B is not pressed, or when C is pressed but B is not pressed
A7 / Repeat the above for the following system. The light is on in three cases: When A is pressed and B is pressed but C is not pressed, or when A is pressed but both B and C are not pressed, or when both A and B are pressed but C is not pressed.
Original Material. Find a system of your choice with three or more inputs and apply the above procedure to generating a Boolean expression. You could think of a home security system looking at doors and pressure pads etc., or else a system in your car.