1. The Bridge Between Symbolic Logic And Electronic Digital Computing
Logic Gates
The Bridge Between Symbolic Logic And Electronic Digital Computing
In this lecture, you will be introduced to digital logic circuit design, beginning with a discussion of logic gates, which are the basic building blocks for logic circuits, and continuing through some simple example circuits. We begin by introducing some basic logic functions that should be provided in order to design logic circuits conveniently.

2. Copyright Gene A. Tagliarini
Basic Logic Functions P Q
P and Q
P or
P nand Q
P nor Q
P xor Q
P xnor Q
not P 1
Logic functions are typically specified using a table that is analogous to a truth table that one might find used in symbolic logic. In the logic table shown, the first two columns represent the possible combinations of values of the independent, binary logic variables P and Q. Each of the next seven columns explicitly specifies the value generated by some combination of P and Q or by inverting P. The logic functions are and, or, nand, nor, xor, xnor, and not. While the behaviors associated with these functions can be memorized by remembering the table, there are some observations that can simplify the process. The logic function:
P and Q generates a 1 if and only if both P and Q have the value 1. If either P or Q has the value 0, the logic function P and Q will have the value 0.
P or Q generates a 0 if and only if both P and Q have the value 0. If either P or Q has the value 1, the logic function P or Q will have the value 1.
P nand Q generates a 0 if and only if both P and Q have the value 1. If either P or Q has the value 0, the logic function P or Q will have the value 1. Notice that P nand Q is P and Q inverted.
P nor Q generates a 1 if and only if both P and Q have the value 0. If either P or Q has the value 1, the logic function P nor Q will have the value 0. Notice that P nor Q is P or Q inverted.
P xor Q generates a 1 if and only if both P and Q have opposite logic values. If P and Q have the same logic value, the logic function P xor Q will have the value 0.
P xnor Q generates a 0 if and only if both P and Q have opposite logic values. If P and Q have the same logic value, the logic function P xnor Q will have the value 0. Notice that P xnor Q is P xor Q inverted.
not P generates a 0 if and only if both P has the value 1. Thus, not P is P inverted.
It is interesting to notice how many distinct logic functions of two variables exist. Since there are two logic variables, there are four possible combinations of the logic values of the independent variables P and Q, corresponding to the four rows of the logic table. The four rows arise from the fact that there are two logic variables and each can be assigned a logic value in one of two ways; hence, there are 2x2=4 (or 2^2) rows in the logic table. In general, there are 2^n rows in a logic table for a logic function of n independent logic variables.
Now, since there are four rows in the table, there are four values that must be assigned in order to fully define a logic function, and since each of those values can be assigned in one of two ways, the fundamental counting theorem guarantees that there are 2x2x2x2=16 possible assignments. Thus, there are nine logic funtions of two variables that have not been specified by the table. Can you specify them? … In general. There are 2^2^n logic functions of n independent logic variables.
Copyright Gene A. Tagliarini
11/25/2018

3. Copyright Gene A. Tagliarini
Logic Gate Symbols
AND OR
NAND
NOR
XOR
XNOR
NOT
Electrical engineers have developed circuit symbols for the logic gates that perform the basic logic functions you will study. Beginning at the top left, we see the symbol for an AND gate. An AND gate will perform the logic function AND on its inputs, which are shown here on the left-hand side of the AND gate symbol, and produce its output a the wire position located on the curved side of the symbol (shown here on the right side). An OR gate is represented using an arrowhead-shaped symbol, with its inputs where the shank of the arrow would be attached and its output at the point of the arrowhead. The OR gate symbol is shown at the top right.
Next notice the symbol at the bottom. The triangle and open circle represent an inverter which performs the logic function NOT. Sometimes, when the function of an inverter is combined with that of another logic function, the triangle may be omitted and the open circle is used by itself. This is precisely the case in the second row of symbols, which indicates a NAND gate (an inverted AND) and a NOR gate (inverted OR).
The circuit symbol for the logic function XOR is the symbol for an OR gate plus an extra arc that cuts across the inputs. The circuit symbol for XOR is shown on the left in the third row. As you would, no doubt guess, the other circuit symbol in row three is for the logic function XNOR.
Copyright Gene A. Tagliarini
11/25/2018

4. Other Useful Circuit Symbols
Binary Probe
Binary Switch
Input Port
Output Port
While there are numerous other useful circuit symbols, the binary probe, the binary switch, the input port, and the output port will be used immediately. The binary probe allows the circuit designer to observe the binary status of a single logic value. Probes will be used to determine the exact values of input values as well as whether or not the output of a circuit is correct.
The binary switch (see the LogicWorks4 display) allows the circuit designer to set the logic value that will be applied to a circuit. By manually switching these values and probing the state of circuit outputs, the designer can determine if the circuit is designed in a logically correct form.
While the basic gates provide a powerful and functionally complete set of building blocks, the circuit designer will soon realize that some higher-level functions occur so frequently that they ought to be incorporated into circuit packages themselves. When the circuit designer recognizes such a circuit, he must specify the inputs and outputs to the circuit to be encapsulated. Input and output ports are used to identify these paths so that more potent building blocks can be developed.
These symbols appear in LogicWorks4 (as well as other schematic design tools) and will be used frequently. You should learn to recognize the symbols and locate them rapidly as you learn to use LogicWorks4.
Copyright Gene A. Tagliarini
11/25/2018

5. Simple Example Circuits
Basic Gates
Input from binary switches
Output read using a binary probe
The last step in this lecture uses LogicWorks4 to demonstrate the behaviors of the basic logic gates and to introduce the LogicWorks4 circuit design tool.
First launch LogicWorks4. After LogicWorks4 starts, use the component selector (on the right-hand side of the circuit design area to select (by double-clicking) an AND-2 gate. This is an AND gate with two inputs, which we will refer to only as an AND gate in this discussion. Place the AND gate on the design area by moving the mouse pointer to the area and clicking the left mouse button. Click the right mouse button to avoid placing any more AND gates.
Now select the binary probe. You can accelerate your search by typing “binary probe” in the area marked “Filter” on the component selector. Select the binary probe from the list by double-clicking and place it on the circuit design pad so that its input touches the output of the AND gate. Notice that the label changes from “z” to “x” when you connect to the gate. The probe was “disconnected” from the circuit but when it was connected, there was not enough information present in the circuit to determine an output value for the AND gate.
Now select the binary switch and place two binary switches on the left-hand side of the AND gate. Left click on the narrow + button to obtain the ability to draw wires and then connect the output of one binary switch to one of the inputs to the AND gate. What symbol does the probe display? Select the arrow (selector) button, then place the pointer on the connected binary switch. When you click the left mouse button, the switch should toggle between 0 and 1. Notice that, if the switch is set at 1 the probe displays “x”. Why? The “x” is displayed because the output of an AND gate cannot be determined by knowing that one of its inputs has logic value 1. HOWEVER, when you toggle the binary switch so that its value is 0, notice that the output of the AND gate immediately becomes 0. Why? The output of the AND gate becomes 0 because the AND logic function always produces a 0 whenever any one of the input values is 0. Thus, when the connected input has logic value, it does not matter what the other input’s logic value may be, the output is 0. Connect the second binary switch to the other input of the AND gate and note that the behavior complies with that specified in the logic table.
When you have finished exploring the AND gate, investigate the behaviors of the other six common gates that were specified in the logic function table.
Copyright Gene A. Tagliarini
11/25/2018

6. Simple Example Circuits
A Steering Circuit
Two input sources
Only one can be selected for output
Output should match the selected input
Finally, consider the design of a steering circuit that will take one bit of data from one or the other of two distinct sources A or B and present the logic value associated with that source as the circuit’s output. Suppose a binary switch S will be used to select which source of data will be output. If the logic value output from S is 1, then the logic value of A will be output. If the logic value output from S is 0, then the logic value of B will be output. To form this circuit begin by using the “lightning bolt” icon to erase all the objects on the design pad. Follow these steps:
1. Place two, two-input AND gates near the middle of the design area, one about one inch above the other.
2. Place two binary switches on the pad and connect each one to the upper input of a different AND gate. These are the switches that provide the signal sources A and B.
3. Place an inverter to the left of the and gates and switches. Connect the output of the inverter to the remaining input of the lower AND gate. Connect the inverter’s input to the lower input of the upper AND gate.
4. Place a binary switch to the left of the inverter and connect the output of the switch to the input of the inverter. This is the selector switch S.
5. Place a two-input OR gate (OR-2) to the right of the AND gates and connect the output of each AND gate to a different input to the OR gate.
6. Place a binary probe to the right of the OR gate and connect the output of the OR gate to the input of the binary probe.
You now have a steering circuit, also known as a multiplexer, a device which enables two (or more) signal sources, at different times, to share a common output path. By changing the values of the switch S (connected to the inverter input) and then toggling the switches A and B, you can observe that the output value tracks the value of either switch A or switch B depending upon whether switch S is set to 1 or 0, respectively. For greater clarity of understanding, you may wish to add binary probes to monitor the outputs of the and gates as well as the inverter.
Copyright Gene A. Tagliarini
11/25/2018