Review Sheet for Final
General Information. As I specified in class, Monday Dec. 5, 2011, the test will be
selected from among the problems that I specify below. Be aware, though, I reserve the
right and duty to ask additional problems that are in the penumbra of these.
1. Be able to define the union and intersection of sets. Be able to define P implies Q. Be
able to define relation, function, injective function, and surjective function. Know the
meanings of domain and range.
2. Be able to define the power set of a set. Letting |Y | denote the number of elements
in a finite set, be able to prove (by induction) that |P(X)| = 2|X| . Be able to list the
elements of a power set for a small set.
3. Be able to prove that if n is not prime, then 2n − 1 is not prime.
Ax+B
for specific values of A, B, C, and
4. Be able to find the inverse function for y = Cx+D
D. Be able to verify that the resulting formula that you found is the inverse. See page
152, for example.
5. Be able to negate complicated expressions such as
(a) ∀ > 0, ∃δ > 0 such that |x − x0 | < δ implies that |f (x) − f (x0 )| < .
(b) ∀ > 0, ∃δ > 0 such that ∀x, we have that |x−x0 | < δ implies that |f (x)−f (x0 )| <
.
6. Be able to complete the square of y = Ax2 + Bx = C to prove the quadratic formula.
≡
7. Suppose that n is a non-negative integer. Thus n ∈ Z+ . Define a relation Z ←n Z by
the formula a ≡n b if and only if n|(b − a). That is, there is an integer q such that
(b − a) = nq.
(a) Show that ≡n is a reflexive relation.
(b) Show that ≡n is a symmetric relation.
(c) Show that ≡n is a transitive relation.
(d) A partition of a set X is a family F of subsets of X such that
i. X = ∪A∈F A,
ii. if A, B ∈ F and A 6= B, then A ∩ B = ∅.
iii. if A ∈ F, then A 6= ∅.
Let j ∈ Z. Let Aj = {a ∈ Z : a ≡n j}. Show that F = {Aj : j ∈ {0, 1, 2, . . . , n −
1}} is a partition of Z. The sets Aj are called equivalence classes.
(e) Show that, as sets, Ai = Aj if and only if i ≡n j. The set Aj will be denoted by
[j] and is called an equivalence class. The element j ∈ Z is called a representative
of the equivalence class.
1
(f) Suppose that n = 5. List the elements of the equivalence classes [0],[1],[2],[3], and
[4].
(g) Let n return to denoting an arbitrary integer. Define [i] ⊕ [j] = [i + j]. You must
show that the operation ⊕ is well-defined. That is, you must show that if [i] = [i0 ]
and [j] = [j 0 ], then [i + j] = [i0 + j 0 ].
(h) Similarly, define [i] ⊗ [j] = [i × j]. Show that the operation ⊗ is well-defined in
the sense that that if [i] = [i0 ] and [j] = [j 0 ], then [i × j] = [i0 × j 0 ].
Please note, the symbols ⊕ and ⊗ are only temporary symbols used for these
problems and have different standard usages within the rest of the mathematical
literature.
(i) Show that [0] serves as the additive identity for ⊕. That is, show that ∀j ∈ Z, we
have [0] + [j] = [j].
(j) Show that [1] serves as the multiplicative identity for ⊗. That is, show that ∀j ∈ Z,
we have [1] ⊗ [j] = [j].
(k) Construct an addition table and a multiplication table for n = 5.
(l) Construct an addition table and a multiplication table for n = 6.
(m) Looking at the multiplication tables for n = 5 and n = 6, what significant difference do you see among the results.
8. Given that 0! = 1, and for 0 < n, we define n! = n(n − 1)!, be able to compute 1!, 2!,
3!, 4!, and 5!. Be able to prove that
(n − 1)!
(n − 1)!
n!
=
+
.
k!(n − k)!
(k − 1)!(n − k + 1)! (k)!(n − k)!
n!
be able to state and prove the binomial theorem. Be able
Given that nk = k!(n−k)!
to give a combinatorial proof that nk is the number of ways of choosing a k element
subset from an n element set.
9. Be able to prove (by induction) that
n−1
X
An − B n = (A − B)
!
Ak B n−k
.
k=0
10. Be able to prove
(a)
1 + 2 + ··· + n =
n
X
k=
k=1
n(n + 1)
2
(b)
2
2
2
1 + 2 + ··· + n =
n
X
k=1
2
k2 =
n(n + 1)(2n + 1)
6
(c)
n
X
n(n + 1)
1 + 2 + ··· + n =
k =
2
k=1
3
3
3
3
3
2