Math 2200: Discrete Math
Spring 2014
Exam 2 Information
• The test will be on Monday, March 24 during the regular class period.
• No calculators. (You won’t need one anyway.)
• You must show your work. No credit will be given if no work is shown. Partial credit will be given for
partially correct solutions.
• You may bring a 3 × 5 index card with notes.
• The test will cover all material from chapters 2, 9, and 4. Questions are likely to be similar to homework
problems or problems on this review.
Important vocab:
(You don’t need to be able to write a definition for all of these, but you should be comfortable working with
them.)
Chapter 2:
Sets
set
element
set builder notation
empty set
universal set
set equality
subset
powerset
union
intersection
complement
difference
function
domain
codomain
range
image
preimage
onto
one-to-one
one-to-one correspondence
injective
surjective
bijective
composition
inverse
Chapter 9:
Relations
relation from A to B
relation on A
reflexive
symmetric
antisymmetric
transitive
equivalence relation
equivalence class
partition
partial order
poset
comparable
total order
maximal/greatest element
minimal/least element
Chapter 4:
Number Theory
a divides b
congruence modulo m
modular arithmetic
division algorithm
a mod b
a div b
greatest common divisor
least common multiple
Euclidean algorithm
fundamental theorem of arithmetic
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Some common kinds of proofs:
• A⊆B
• A = B (sets)
• A function f is one-to-one/onto/a one-to-one correspondence
• A relation R is reflexive/symmetric/anitsymmetric/transitive
Examples of these kinds of proofs:
• Let A be the set of numbers divisible by 6 and B the set of numbers divisible by 3. Show that A ⊆ B.
Show that A 6= B.
• Let
A = {n ∈ Z|n is odd}
B = {n ∈ Z|n mod 4 = 1}
C = {n ∈ Z|n mod 4 = 3}
Show that A = B ∪ C.
• Let f : R → Z be the function that maps each real number x to the largest integer y such that y ≤ x.
Is f onto? Is f one-to-one? (Prove your answers!)
• Let f : Z → Z be defined by f (n) = 3n + 2. Is f onto? Is f one-to-one? (Prove your answers!)
• Let ∼ denote the relation on Z × Z given by (a, b) ∼ (c, d) if there is some k ∈ Q such that a = kc and
b = kd. (This is the same as saying ab = dc ). Show that ∼ is an equivalence relation.
• By the fundamental theorem of arithmetic, any integer n with a ≥ 2 can be written uniquely in the
form a = 2n k for an odd number k. Define a relation on the set {z ∈ Z|z ≥ 2} by 2n k 2m ` if k
and ` are odd numbers and n ≤ m. Show that is a partial order.
You should also review your homework exercises.
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