MATH 111 COMMENTS ON HW4
10. (a) False: A−Bconsists of the elements of Awhich are not in B.B−A
is the set of elements of Bwhich are not in A. Quite different! For
example, if A={0,1,2,3}and B={2,3,4,5}, then A−B={0,1}
and B−A={4,5}.
There were two check marks for this problem: one for correctly saying
that the statement is false, and another for giving a correct reason or
example. Many of you gave explanations along the lines of “If A= 2
and B= 1, then 2 −16= 1 −2”. There are several problems with this.
First, in this exercise, Aand Bare given to be sets, not numbers. You
may have meant to compare n(A−B) and n(B−A), the sizes of A
and B, but then you must be cautious: n(A−B) is not necessarily
equal to n(A)−n(B)—only when Bis a subset of A. (And of course
no set has a size which is negative!)
(c) True: A∪ ∅ =A, and certainly A⊆A.
(e) False: the empty set has only one subset—the empty set! So the empty
set is equivalent to all of its subsets. (The empty set is the only set
with this property. Indeed, the empty set is a subset of every set, so if
a set is equivalent to all of its subset, it is therefore equivalent to the
empty set.)
11. A common incorrect answer was 3, which is the number of students who like
calculus and algebra but not geometry. The problem asks for all students
who like calculus and algebra (whether or not they like geometry), so the
answer is 3 + 4 = 7.
12. (a) Deceptive! While this expression looks like an expression where you’d
use the distributive property, the property that is actually used is
commutativity of addition.
(c) The best answer here is: the identity property of multiplication. The
commutative property can be used to obtain 1 ·14 = 14 ·1, but the
fact that these are both equal to 14 is the identity property. However,
I gave a check mark for giving commutativity as the answer.
16. In hindsight, this problem was ambiguous: one could interpret the state-
ment as saying that Jim received one $20 raise after the sixth month, or
as saying that there was a $20 raise in each of the final six months. Both
interpretations were accepted as correct.
17. It’s not enough to say “nothing can be divided by zero”: why can’t anything
be divided by zero? We want to way that a/b =qif qis the unique number
such that a=b·q. If we try to divide, say, 1 by 0, the problem is that there
