Definitions

Question 1

What is an ordered set?

Answer 1

An ordered set is a set S in which an order is defined.

Question 2

Provide an example of an ordered set.

Answer 2

l is an ordered set if r < s is defined to mean that s - r is a positive rational number.

Question 3

What is the definition of least upper bound?

Answer 3

The least upper bound of a set E, denoted as sup E, is an element a in S such that:
* a is an upper bound of E
* If y < a, then y is not an upper bound of E.

Question 4

What does the notation a = sup E represent?

Answer 4

It indicates that a is the least upper bound of the set E.

Question 5

What is the definition of greatest lower bound?

Answer 5

The greatest lower bound of a set E, denoted as inf E, is an element a in S such that:
* a is a lower bound of E
* No B with β > a is a lower bound of E.

Question 6

What does the notation a = inf E represent?

Answer 6

It indicates that a is the greatest lower bound of the set E.

Question 7

True or False: A least upper bound is unique.

Answer 7

True

Question 8

Fill in the blank: The least upper bound is also known as the _______.

Answer 8

supremum

Question 9

Fill in the blank: The greatest lower bound is also known as the _______.

Answer 9

infimum

Question 10

What is an order on a set S?

Answer 10

A relation with two properties:
1. For any x, y in S, one and only one of x < y, x = y, or y < x is true.
2. If x < y and y < z, then x < z.

Question 11

True or False: In an ordered set, for any two elements x and y, both x < y and y < x can be true.

Answer 11

False

Question 12

What does the transitive property in an ordered set state?

Answer 12

If x < y and y < z, then x < z.

Question 13

What is a subset?

Answer 13

If A and B are sets, and every element of A is an element of B, then A is a subset of B.

Denoted as A ⊆ B or B > A.

Question 14

How is a proper subset defined?

Answer 14

A is a proper subset of B if A is a subset of B and there exists an element in B that is not in A.

Denoted as A ⊂ B.

Question 15

What notation is used to express that A is a subset of B?

Answer 15

A ⊆ B or B > A.

These symbols indicate the relationship between the two sets.

Question 16

What is true for every set A regarding subsets?

Answer 16

A ⊆ A.

Every set is a subset of itself.

Question 17

What does it mean for a set E to be bounded above?

Answer 17

There exists a B in S such that x ≤ B for every x in E.

Question 18

What is an upper bound?

Answer 18

An element B in S that satisfies x ≤ B for every x in E.

Question 19

How are lower bounds defined?

Answer 19

By replacing ≤ with ≥ in the definition of upper bounds.

Question 20

Fill in the blank: If there exists a B in S such that x ≤ B for every x in E, we say that E is _______.

Answer 20

bounded above

Question 21

What is the least-upper-bound property?

Answer 21

An ordered set S has the least-upper-bound property if for any non-empty subset E of S that is bounded above, the supremum (sup E) exists in S.

Question 22

True or False: The least-upper-bound property requires that every non-empty subset of an ordered set has a supremum.

Answer 22

False

Question 23

Fill in the blank: An ordered set S is said to have the least-upper-bound property if for any non-empty subset E of S that is bounded above, _______ exists in S.

Answer 23

sup E

Question 24

What does it mean for a set E to be bounded above?

Answer 24

A set E is bounded above if there exists a number that is greater than or equal to every element in E.

Question 25

What is an example of a set that does not have the least-upper-bound property?

Answer 25

The set of rational numbers Q.

Question 26

What is the significance of the least-upper-bound property in ordered sets?

Answer 26

It ensures that every non-empty subset that is bounded above has a least upper bound within the set.

Question 27

What is the definition of a field?

Answer 27

A field is a set F with two operations, called addition and multiplication, which satisfy the field axioms (A), (M), and (D).

Question 28

What are the two operations defined in a field?

Answer 28

Addition and multiplication.

Question 29

What is a function from set A to set B?

Answer 29

A function f is a mapping from A to B such that each element x in A is associated with an element f(x) in B.

Question 30

What is the domain of a function f?

Answer 30

The domain of f is the set A, where f is defined.

Question 31

What are the values of a function f?

Answer 31

The values of f are the elements f(x) for each x in the domain A.

Question 32

What is the range of a function f?

Answer 32

The range of f is the set of all values f(x) obtained from the domain A.

Question 33

Fill in the blank: The set A is called the _______ of f.

Answer 33

domain

Question 34

Fill in the blank: The set of all values of f is called the _______.

Answer 34

range

Question 35

What is a function from a set A to a set B?

Answer 35

A function f from A to B is a rule that associates to each element x in A exactly one element f(x) in B. The set A is called the domain of f, and the set of all values f(x) is called the range of f.

Question 36

What is the image of a subset E ⊆ A under a function f?

Answer 36

The image of E under f, denoted f(E), is the set of all elements f(x) where x ∈ E.

Question 37

When do we say that a function f maps A onto B?

Answer 37

A function f maps A onto B if f(A) = B, meaning every element of B is the image of some element in A.

Question 38

What is the inverse image of a subset E ⊆ B under a function f?

Answer 38

The inverse image of E under f, denoted f⁻¹(E), is the set of all x ∈ A such that f(x) ∈ E.

Question 39

What does it mean for a function f to be one-to-one (1-1)?

Answer 39

A function f is one-to-one if, for every y ∈ B, the set f⁻¹(y) contains at most one element of A. Equivalently, f is one-to-one if f(x₁) ≠ f(x₂) whenever x₁ ≠ x₂.