MTH 182
Exam 1 Review Sheet Version 1
Spring 2017
Section 4.4 Recursively Defined Sequences and Strong Induction
• Be familiar with the concept of a recursively defined sequence including how to generate the terms of a
sequence that is defined recursively.
• Know the structure and formal steps of a proof by Strong Mathematical Induction (see page 138 and
page 141).
• Be able to use the first three to five elements of a recursively defined sequence to make a reasonable
conjecture of the explicit closed-form formula for the sequence. Review Section 4.4 Problems 1, 5, and 7.
• Review Results 4.29, 4.30, 4.32, 4.33, 4.34, and 4.35 as well as Example 4.31.
• Review Section 4.4 Problem 10 and Quiz 1.
Section 5.1 Relations
• Be familiar with the definition of a relation R on a set S. Review Example 5.7.
A×B
• Know that the number of possible relations from a set A to a set B is ! 2
(see Class notes).
• Know the definitions of the reflexive, symmetric and transitive properties of a relation on a set. Review Examples 5.10 –
5.13 and Quiz .
Section 5.2 Equivalence Relations
• Know the definition of an equivalence relation (see page 152) and know how to determine whether or not a relation is
an equivalence relation including those defined by a formulas. Review Examples 5.15 – 5.16 and Problems 1-2 of
Worksheet 01 and Problems 1 and 2 of Quiz 2.
• Know the definition of an equivalence class ! [a] of ! a ∈S with respect to an equivalence relation R on S (see page 153).
• Be aware that ! a ∈[ a ] ⊂ S , hence, ! S =
∪[ a ] .
Review Examples 5.18 – 5.21 and Worksheet 01 and Quiz 2.
a∈S
• Be aware that if ! [ a ], [ b ] are equivalence classes of a relation R, then (1) ! aRb ⇔ [ a ] = [ b ] and (2) either ! [ a ] = [ b ] or
! [ a ] ∩ [ b ] = ∅ . See Theorem 5.22 and Corollary 5.24.
• Be familiar with the concept of a partition of a set S (see Section 2.4) and know that the set ℘
! of distinct equivalence
classes of an equivalence relation R on a set S forms a partition of S (see Theorem 5.23).
• Know that each partition !℘ = { P1 , P2 , P3 ,!} of a set S corresponds to a unique equivalence relation R on a S; namely,
the one defined by ! aRb if and only if ! a,b belong to the same ! Pj ∈℘ . Thus, finding all the partitions of a set S is
equivalent to finding all the equivalence relations on S (see class notes).
• Know that the number of partitions of the set S equals the number of equivalence relations on S and how to construct
the equivalence relation R from a partition of a finite set S (see the class notes and Problem 3 of Worksheet 01).
Section 5.3 Functions
• Know the definition of a function from a set A and a set B including how it is different from other kinds of relations
from A to B (see page 159). Review Example 5.27 and Problem 1 of Worksheet 02.
• Be familiar with the terms domain, codomain and the range of a function.
• Know how to represent a function from a finite set A to a finite set B by an arrow diagram (see Example 5.30).
A
• Know how to find all the functions from a finite set A to a finite set B and that there are a total of ! B functions from A
to B (see class notes).
• Know how to tell if a graph of a relation is a function by using the vertical line test. See class notes and
Examples 5.22 – 5.23.
• Be familiar the definitions of the identity function 1A : A → A . See Example 5.35.
MTH 182
Exam 1 Review Sheet Version 1
Spring 2017
Section 5.3 Functions (Continued)
• Know how to form the composition of two functions f : A → B and g : B → C including how to represent the
composition a set of ordered-pairs. Be aware that the composition ! g ! f is evaluated from right to left while the
f
g
→ B→C
diagram ! A
of the composition is drawn left to right! See Examples 5.42 – 5.44 and Problems 28 – 31.
• Let function f : A → B be a function. Know the meaning of the term image of an element under f and the image of
subset X ⊆ A under f. See Example 5.27, Problem 3 and Problem 7.
Section 5.4 Bijective Functions
• Know the meaning of a injective (or one-to-one) function. Review Examples 5.46 – 5.47. • Know how determine whether or not a function is injective. If a function is injective, know how to prove that it is. If it is not injective, know how to provide an example to show that it isn’t. Review Examples 5.48 – 5.50. • Know how to compute the number of injective functions f : A → B when A and B are finite sets. See class notes. • Know the meaning of a surjective (or onto) function. Review Examples 5.52 – 5.54. • Be aware of how the domain and codomain of a function affects whether or not a function is injective or is surjective. Review Result 5.53, Example 5.55, and class notes. • Know the meaning of a bijective (or one-to-one correspondence) function and how to determine whether or not a function is bijective (including functions between finite sets). Review Examples 5.58 – 5.62.
• Know that a function ! f : A → B has an inverse ! g : B → A if and only if f is a bijection. Also, be aware that that the
! g : B → A is an inverse for f if and only if ! g ! f = 1A and! g ! f = 1B , and that g is defined by the condition
! g(b) = a if ! f (a) = b . In terms of ordered-pairs, this means that ! (b, a) ∈g if and only if ! (a, b) ∈ f .
Remarks
1. Be sure to review Quizzes 1 – 2 , Worksheets 1-2 and the assigned problems.
2. I won’t ask questions about common functions (pages 162 – 165) on this exam!
Exam I
MTH 182 Friday, February 24, 2017
Good Luck! Get some sleep before the exam!