- One times eight gives you 8
- Zero times sixteen gives you 0
- One times thirty-two gives you 32
The decimal-numeral equivalent is therefore 1 + 2 + 0 + 8 + 0 + 32 = 43. This is the
quantity you commonly imagine as forty-three.
- There are at least three ways to solve this problem. You can “cheat” and use Table 1-2.
To do it manually, note that forty-three, the decimal equivalent as derived in the
solution to Prob. 8, is equal to five times eight, plus three more. That means you can
define quantity forty-three as the octal numeral 53. You can also add up the values in
octal arithmetic. That’s tricky and hasn’t been covered in this chapter, so we won’t show
it here. - As with the decimal-to-octal conversion, there are at least three ways to solve this
problem. You can “cheat” and use Table 1-2. To do it manually, note that forty-three
is equal to two times sixteen, plus eleven more. In hexadecimal notation, eleven is
represented by B. That means you can define the quantity forty-three in hexadecimal
form as the numeral 2B. The last method is to add up the values in hexadecimal
arithmetic. Again, that’s tricky and hasn’t been discussed, so we won’t do it here.
Chapter 2
- The null set is a subset of any set. The null set lacks elements, like an empty bank
account lacks money. You can say that it contains nothing as an element! If you have
a certain set A with known elements, you can add nothing, and you always end up
with the same set A. The null set is a subset of itself, although not a proper subset of
itself. Look at an example. If you let the written word “nothing” actually stand for
nothing, then
∅= { } = {nothing}
and
{nothing}⊆ {nothing, 1, 2, 3}
so therefore
∅ ⊆ {1, 2, 3}
Keep in mind that a subset is not the same thing as a set element. The null set contains
nothing, but the null set is not itself nothing. An empty bank account is a perfectly valid
account unless somebody closes it.
- You can build up an infinite number of sets if you start out with nothing. First, take
nothing and make it an element of a set. That gives you the null set. Then consider the
set containing the null set, that is, {∅}. This is a legitimate mathematical thing, but
Chapter 2 589