Last updated: 2/19/03
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Worksheet: Truth Tables and Switch Circuits 
Computers and Society, GST 2710

Truth Tables
A Truth Table shows how the truth or falsehood of a result depends upon the starting values. In Philosophy and Logic, truth tables are used to analyze statements such as, "All men are mortal. Socrates is a man. Therefore Socrates is mortal." Truth Tables apply to computer science because a bit can be only 0 or 1, just like a statement such as, "All men are mortal" can only be false or true. The usual convention is 0 corresponds to False while 1 corresponds to True. This is what we wil use here, although we could just as well choose the opposite. We will first apply Truth Tables to binary multiplication and addition. Later we will use electric switches as basic devices or building blocks for computers, and then more abstract devices known as Logic Gates. In each case, we will fill out Truth Tables, and then use these Truth Tables to see if such devices correspond to binary arithmetic. 

Truth Table for binary multiplication:

Consider bits A, B and C, where C = A × B.

This Truth Table summarizes everything about binary multiplication.

Next, consider bits A, B, C and D where C = units bit of A + B and D is the carry bit.

Switches (there will be a connection)

We start with a simple electrical circuit with a battery, a switch A and a light bulb C, as in the figure above. The switch "A" shown here in a "neutral" position, but really cannot stay there except when moving from one position to the other.  The switch has a pivot or joint (the little circle at the left), two contacts (the upper and lower little circles on the right) and a switch element (the straight line, which pivots at the joint and is in contact with either the upper or lower contact).

With the circuit shown, if the switch is up, there is a complete path for the current, the current flows, and the bulb is on. Test the existence of a complete circuit by starting out from the "+" battery terminal - the one with the little bump on the end, and tracing around with your pencil, without picking the pencil up, and moving only along lines or across the little circles that are the switch contacts. The switch provides a current path from the pivot to whichever contact the switching element is touching. Electricity, and your pencil, can pass right through the pivot and contacts, but cannot jump between them - it can only take the path laid out by the switching element. If you can find a complete path with no gaps, then there is a complete circuit. When there is a complete circuit, current flows. when current flows the bulb is light. See the shining light marks around the bulb?

For the circuit shown, if the switch is down, there is no path for the current, current cannot flow, and the bulb is off. Here, there is a gap when your pencil reaches the end of the down-pointing switch; there is no path from there to the rest of the circuit on the right side.

In binary notation, if switch A is down, we will call that a zero; up and we will call it one. If bulb C is off, we will call that zero; on and we will call it one. We can now turn this into a Truth Table.

Fill out the truth table for the switch combination below. Now there are two switches, A and B, and either one can be up (1) or down (0), for a total of four combinations. Frequently there are switches at the top and bottom of a set of stairs, wired in this parallel combination. If the light is off, flipping either switch turns the light on. Similarly, if it is on, flipping either switch turns it off. NOTE: In this style of electrical diagram, wires cross without making electrical contact. When you are tracing the circuit, and you come to a crossing, you cannot turn but must only continue straight across in your original direction. This is "crossing without contact."

Now fill out the truth table below for binary addition (C = A + B)