Exam 1 Definitions

Question 1

even

Answer 1

divisible by two

Question 2

divisible

Answer 2

if b|a then bc = a

Question 3

odd

Answer 3

a is odd if a = 2x+1

Question 4

prime

Answer 4

p > 1 and divisors of p are 1 and p

Question 5

composite

Answer 5

a is composite if b|a such that 1 < b < a

Question 6

subset

Answer 6

A and B are sets
every element of A is in B
A ⊆ B means A is a subset of B.

Question 7

power set

Answer 7

set of all subsets of A

Question 8

union

Answer 8

A or B
A ∪ B

Question 9

intersection

Answer 9

A and B
A ∩ B

Question 10

theorem about sets

Answer 10

1. A ∪ B = B ∪ A and A ∩ B = B ∩ A (commutative properties)
2. A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C
3. A ∪ Ø = A and A ∩ Ø = Ø
4. A ∪ (B ∩ C) = ( A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). (distributive properties)

Question 11

disjoint

Answer 11

A ∩ B = Ø
there is nothing in both A and B

Question 12

pairwise disjoint

Answer 12

a collection of sets where Ai ∩ Aj = Ø and i ≠ j so that there are no two of the same elements in common

Question 13

set difference

Answer 13

A - B = x
x is in A but not B, denoted A ∆ B = (A - B) ∪ (B - A)

Question 14

cartesian product

Answer 14

A x B, set of ordered pairs by taking a from A and b from B to make (a,b)

Question 15

relation

Answer 15

set of ordered pairs

Question 16

relation on, between sets

Answer 16

R is a relation on A provided R ⊆ A x A
R is a relation between A and B provided R ⊆ A x B

Question 17

inverse relation

Answer 17

reversing the order for all ordered pairs in R

Question 18

properties of relations

Answer 18

• x ∈ A then xRx (reflexive)
• x ∈ A then x ≠ x (irreflexive)
• x, y ∈ A then xRy ⇒ yRx (symmetric)
• x, y ∈ A then (xRy ^ yRx) x = y (antisymmetric)
• x, y, z ∈ A then (xRy ^ yRz) ⇒ xRz (transitive)

Question 19

equivalence relation

Answer 19

the relation is reflexive, symmetric, and transitive

Question 20

congruence modulo n

Answer 20

x ≡ y (mod n) provided n|(x - y)

Question 21

equivalence class

Answer 21

a ∈ A where [a] is the set of all elements of A related by R
[a] = {x ∈ A : xRa}

Question 22

partition

Answer 22

the partition of A has to be nonempty, pairwise disjoint and their union is A

Question 23

function

Answer 23

f = (a, b) and f = (a, c) implying b = c

Question 24

function notation

Answer 24

f(a) exist provided that (a, b) ∈ f
f(a) = b
if there is no ordered pair than f(a) is undefined

Question 25

domain

Answer 25

set of all possible first elements in the ordered pair in f
dom f

Question 26

image

Answer 26

set of all possible second elements in the ordered pair of f
im f

Question 27

function from A to B

Answer 27

dom f = A and im f ⊆ B
(f : A → B)

Question 28

proposition inverse function

Answer 28

if f^-1 then dom f = im f^-1 and f = dom f^-1

Question 29

onto

Answer 29

f is onto B for f(a) = b or im f = B

Question 30

bijection

Answer 30

function is both one-to-one and onto