We can combine two inputs to produce an output using many rules other than AND or OR. We won’t worry about what the switches would look like for the new rules below… we would have to make our simplified model of an electric circuit more complex to see how they work. Instead, we will do what computer scientists normally do - abstract away from the actual electronics and focus on the logical rules they describe.
Subsection5.4.1XOR Gates
The exclusive or gate, known as XOR describes a gate that only outputs a 1 only if one of the inputs is 1 and the other is 0. In other words, we can have either input on, but not both on.
Table5.4.1.Truth table for XOR
Input 1
Input 2
Output
0
0
0
0
1
1
1
0
1
1
1
0
It is the logical rule we need if we want to decide the sign of a multiplication problem. If input 1 represents “the first number negative” and input 2 represents “the second number is negative”, then XOR’s output would tell us “Is the answer negative”. If either one of the two numbers, but not both of them, were negative, then the answer would be negative.
Watch out for XOR and OR - in English we often use the word “or” to mean both things. The statement “Entrées come with fries or a salad” probably means you can get one or the other, but not both. In that statement, or means the XOR rule. On the other hand, imagine a class syllabus that states: “If you turn in copied work, or the wrong assignment, you will get a 0”. You would certainly get a 0 if you turned in a plagiarized copy of the wrong assignment (both parts are true). In this case, or means logical OR - if either part is true, the answer is true.
Subsection5.4.2NOT Gates
The NOT gate is special in that it only has one input. The output is always the opposite of the input:
Table5.4.2.Truth table for NOT
Input
Output
0
1
1
0
The NOT operation is what we use if we have a truth value like “it is raining” but want to have the value for “it is not raining”. Anytime we want to say “We do not have X” we can say “NOT X”.
Because the NOT rule applies to only one input, it can be applied to a single-bit pattern like 0101. To figure out the answer, simply “flip” every bit from 1 to 0 or vice versa:
Table5.4.3.NOT of 0101
Input pattern
1
0
1
0
NOT result
0
1
0
1
Note5.4.4.
People refer to an AND gate followed by a NOT as a NAND (Not-AND). That combination negates what the AND gate produces, producing the exact opposite - a NAND combo will produce an answer of 0 when both inputs are on and 1 in every other case. Similarly, a NOR is an OR followed by a NOT, and an XNOR (or ENOR) is a NOT after an XOR. You are not responsible for memorizing the truth tables for these gates, but if you happen to see them, now you know what they are.
Checkpoint5.4.5.
What is the result of applying NOT to 1110?
Checkpoint5.4.6.
What is the result of applying XOR to the following?
