5.4. XOR and NOT Gates

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.

XOR Gates

The exclusive or gate, known as XOR (or sometimes EOR) describes a gate that only outputs a 1 only if one input is on and the other off. It is the rule we can use to determine if the result of a multiplication is positive or negative. If we think of the inputs as “is the first number negative?” and “is the second number negative?” and the output as “the result is negative”, we can see that the truth table below gives us the correct answer for each of the four possible situations. The only way to get a negative product is to have exactly one negative input (01 or 10), if both inputs are negative (11), the result is positive:

Truth table for XOR
Input 1 Input 2 Output
0 0 0
0 1 1
1 0 1
1 1 0

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 true). In this case, or means logical OR - if either part is true, the answer is true.

NOT Gates

The NOT gate is special in that it only has one input. The output is always the opposite of the input:

Truth table for NOT
Input 1 Output
0 1
1 0

It is the kind of logic we need if we know the answer to “is a certain number greater than 0?” but want to answer the question “is the number less than or equal to 0?”. The answer to the second question is always the opposite of the first question.

For example, if the number in question is -3, the answer to the first question is NO (0) and the answer to the second is YES (1). Even if I do not tell you the number in question, but I do tell you the answer to “is the number greater than 0?” is YES (1), you could determine that the answer to “is it less than or equal to 0” by applying the NOT rule.

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 verse:

NOT of 0101
Input pattern 1 0 1 0
NOT 0 1 0 1

Note

People refer to a 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.

Self Check

What is the result of applying NOT to:
1110?

What is the result of applying XOR to:
0101
0011?