MATH 42 San Jose State University Applied Rational Equations Questions


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1. For the holidays, you are planning to have either a Christmas tree or
no tree in the corner of the living room. If and only if you have a
tree, you have exactly one of 4 different choices of the tree (fir, pine,
banana, and spruce) and exactly one of 3 different types of ornaments
(stars, bananas, or balls). If you do not have a tree, you can get up to
5 boxes (including 0) of presents in the corner of the room. We only
care about the set of objects in the corner of the room, and all other
information not mentioned here (type of present, size of boxes, physical
location, etc.) is considered irrelevant for this problem. How many
different possible configurations are there for decorating the corner of
the room? (as an example, one possible configuration is “a pine tree
with ball decorations” and another is “no tree and 4 boxes”)
2. Suppose V6 is rational (optional: you can prove this look it up if
you want; it is educational but hard if you haven’t seen it). Prove
or disprove: V2 + V3 is rational, (hint: you probably want to prove
something like: “if x is rational and y is rational, ay is rational.”)
3. For this problem, we define sets X = {Alice, Bob, Carol} and Y
{Red, White, Blue},
(a) How many bijective functions are there from X to Y?
(b) How many relations that are not functions are there from X to
(c) Is it possible that there exists an injective function f : X+Y
where the range of f is not equal to its codomain? Why or why
(d) What about a surjective function?
4. Can you make change for 137 dollars with only 3 dollar and 12 dollar
bills in the world? Prove if you can or cannot
5. The law says “if you are not over 21 then you cannot drink.” You work
for a bar and your goal is to prevent people from breaking the law.
You can either check (a) people who are drinking or (b) people who
are not drinking, and then punish them (c) if they are over 21 or (d) if
they are under 21. Which of (a) or (b) are you picking? Which of (c)
or (d) are you picking? Why? (your solution must involve statement
forms and logic, not just “English” deduction)
6. You need to pick a captain and 2 sub-captains (all 3 people must be
different to make a team of 3 from a set of n people. (For example,
if the n = 5 people are Alice, Bob, Carol, DeAngelo, and Egani, one
choice of the team is Carol is the captain and Alice and De Angelo are
co-captains. The order of the co-captains do not matter). How many
ways to do this are there?

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rational number

Injective function

algebraic operations

possible conjugations

number of functions

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