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Boolean Algebra
67 questions By Tony R. Kuphaldt
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Question 4 of 67
The following set of mathematical expressions is the complete set of “times tables” for the Boolean number system:
$$0 × 0 = 0$$
$$0 \ x \ 1=0$$
$$1 \ x \ 0=0$$
$$1 \ x \ 1=1$$
Now, nothing seems unusual at first about this table of expressions, since they appear to be the same as multiplication understood in our normal, everyday system of numbers. However, what is unusual is that these four statements comprise the entire set of rules for Boolean multiplication!
Explain how this can be so, being that there is no statement saying 1 ×2 = 2 or 2 ×3 = 6. Where are all the other numbers besides 0 and 1?
Reveal answerBoolean quantities can only have one out of two possible values: either 0 or 1. There is no such thing as “2” - or any other digit besides 0 or 1 for that matter - in the set of Boolean numbers!
Notes:Some students with background in computers may ask if Boolean is the same as binary. The answer to this very good question is “no.” Binary is simply a numeration system for expressing real numbers, while Boolean is a completely different number system (like integer numbers are too irrational numbers, for example). It is possible to count arbitrarily high in binary, but you can only count as high as “1” in Boolean.
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Question 5 of 67
Boolean algebra is a strange sort of math. For example, the complete set of rules for Boolean addition is as follows:
$$0+0=0$$
$$0+1=1$$
$$1+0=1$$
$$1+1=1$$
Suppose a student saw this for the very first time, and was quite puzzled by it. What would you say to him or her as an explanation for this? How in the world can 1 + 1 = 1 and not 2? And why are there no more rules for Boolean addition? Where is the rule for 1 + 2 or 2 + 2?
Reveal answerBoolean quantities can only have one out of two possible values: either 0 or 1. There is no such thing as “2” in the set of Boolean numbers.
Notes:Boolean algebra is a strange math, indeed. However, once students understand the limited scope of Boolean quantities, the rationale for Boolean rules of arithmetic make sense. 1 + 1 must equal 1, because there is no such thing as “2” in the Boolean world, and the answer certainly can’t be 0.
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Question 6 of 67
Surveying the rules for Boolean addition, the 0 and 1 values seem to resemble the truth table of a very common logic gate. Which type of gate is this, and what does this suggest about the relationship between Boolean addition and logic circuits?
Rules for Boolean addition: $$0+0=0$$
$$0+1=1$$
$$1+0=1$$
$$1+1=1$$Reveal answerThis set of Boolean expressions resembles the truth table for an OR logic gate circuit, suggesting that Boolean addition may symbolize the logical OR function.
Notes:Students need to be able to readily associate fundamental Boolean operations with logic circuits. If they can see the relationship between the “strange” rules of Boolean arithmetic and something they are already familiar with (i.e. truth tables), the association is made much easier.
Question 29, answer 3 has a typo - should be A*B + !(D)*E not A*B+!(D*E)