Chapter 1 Speaking Mathematically

x R y means that

(x, y) ∈ R. x is related to y by R

proper subset A⊂B

A is a subset of B, but A is not equal to B {9,14} ⊂ {9,14,28}

subset A⊆B

A is a subset of B. set A is included in set B. {9,14,28} ⊆ {9,14,28}

proper superset A⊃B

A is a superset of B, but B is not equal to A. {9,14,28} ⊃ {9,14}

superset A⊇B

A is a superset of B. set A includes set B {9,14,28} ⊇ {9,14,28}

Let A = {1, 2, 3}, B = {3, 1, 2}, and C = {1, 1, 2, 3, 3, 3}. What are the elements of A, B, and C? How are A, B, and C related?

A, B, and C have exactly the same three elements: 1, 2, and 3. Therefore, A, B, and C are simply different ways to represent the same set.

What does A ⊆ B mean?

For all elements x, if x ∈ A then x ∈ B

What is the first element of (1, 1)?

In the ordered pair (1, 1), the first and the second elements are both 1.

Is (1, 2) = (2, 1)?

No. By definition of equality of ordered pairs, (1, 2) = (2.1) if, and only if, 1 = 2 and 2 = 1. But 1 ≠ 2, and so the ordered pairs are not equal.

Which of the following are true statements? a. 2 ∈ {1, 2, 3} b. {2} ∈ {1, 2, 3} c. 2 ⊆ {1, 2, 3} d. {2} ⊆ {1, 2, 3} e. {2} ⊆ {{1}, {2}} f. {2} ∈ {{1}, {2}}

Only (a), (d), and (f) are true.

How many elements are in the set {1, {1}}?

The set {1, {1}} has two elements: 1 and the set whose only element is 1.

What does A⊄B mean?

There is at least one element x such that x ∈ A and x ∉B.

For each nonnegative integer n, let Un = {n,−n}. Find U1, U2, and U0.

U1 = {1,−1}, U2 = {2,−2}, U0 = {0,−0} = {0, 0} = {0}.

Given sets A and B, a relation from A to B is ______ .

a subset of the Cartesian product A × B

A function F from A to B is a relation from A to B that satisfies the following two properties: a. for every element x of A, there is _____ . b. for all elements x in A and y and z in B, if then ____ .

a. an element y of B such that (x, y) ∈ F (i.e., such that x is related to y by F) b.(x, y) ∈ F and (x, z) ∈ F; y = z

power set 2^A or P(A)

all subsets of A

complement A^c

all the objects that do not belong to set A

equality A=B

both sets have the same members A={3,9,14}, B={3,9,14}, A=B

ordered pair (a,b)

collection of 2 elements

When the elements of a set are given using the set-roster notation, the order in which they are listed_________ .

does not matter

For a set A to be a subset of a set B means that, .

every element in A is an element in B

A conditional statement asserts that if one thing ______ then some other thing ______.

is true; also has to be true

not element of x∉A

no set membership A={3,9,14}, 1 ∉ A

relative complement A\B or A-B

objects that belong to A and not to B A = {3,9,14}, B = {1,2,3}, A \ B = {9,14}

symmetric difference A∆B or A⊖B

objects that belong to A or B but not to their intersection A = {3,9,14}, B = {1,2,3}, A ∆ B = {1,2,9,14}

intersection A∩B

objects that belong to set A and set B ex. A ∩ B = {9,14}

union A∪B

objects that belong to set A or set B ex. A ∪ B = {3,7,9,14,28}

not subset A⊄B

set A is not a subset of set B {9,66} ⊄ {9,14,28}

not superset A⊅B

set A is not a superset of set B {9,14,28} ⊅ {9,66}

element of a∈A

set membership A={3,9,14}, 3 ∈ A

Symbol Z

set of all integers

cartesian product A×B

set of all ordered pairs from A and B

universal set U

set of all possible values

Symbol Q

set of all rational numbers, or quotients of integers

Symbol R

set of all real numbers

Set Builder notation what does this mean? {x ∈ S | P(x)}

the set of all elements x in S such that P(x) is true

The symbol Z denotes

the set of all integers

Given sets A and B, the Cartesian product A × B is .

the set of all ordered pairs (a,b) where a is in A and b is in B

The symbol Q denotes .

the set of all rational numbers

The symbol R denotes .

the set of all real numbers

The notation {x | P(x)} is read .

the set of all x such that P(x)

If F is a function from A to B and x is an element of A, then F(x) is .

the unique element of B that is related to x by F

Given a property that may or may not be true, an existential statement asserts that ________for which the property is true.

there is at least one thing

A universal statement asserts that certain property is _________ for ____________

true; all elements of a set

Is {0} = 0?

{0} not equal to 0 because {0} is a set with one element, namely 0, whereas 0 is just the symbol that represents the number zero.

empty set Ø

Ø = {}