Let A be a well-ordered set. Then If B is similar to A, then B is well-ordered.

• A) Given statement is true.
• B) Given statement is false.
• C)
• D)

Let S(A) denote the collection of all initial segments of elements in a well-ordered set A, and let S(A) be ordered by set inclusion. Then A is similar to S(A) and, in particular, the function f : A →S(A) defined by f(x) = s(x) is a similarity mapping between A and S(A).

• A) Given statement is false.
• B) Given statement is true.
• C)
• D)

Consider the usual ordering ≤ of the positive integers P = {1, 2, 3,…} and the negative integers A = {-1, -2, -3, …}. Then P is not order-isomorphic to A.

• A) False
• B) True
• C)
• D)

Consider a collection C of well-ordered sets. Each well-ordered set A in C is assigned a symbol in such a way that any two well-ordered sets A and B are assigned the same symbol if and only if A and B are similar. This symbol is called the ordinal number of A.

• A) Given statement is false.
• B) Given statement is true.
• C)
• D)

The ordinal number of set {1,2,3} is _____________.

• A) 3
• B) 1
• C) None of these.
• D) 2

Suppose A and B are well-ordered sets, and suppose A is similar to an initial segment of B. Then A is said to be _________ than B.

• A) longer
• B) shorter
• C)
• D)

The set of positive integers P = {1, 2, 3, …} is not order-isomorphic to the set of even positive integers E = {2, 4, 6, …}.

• A) True
• B) False
• C)
• D)

Two ordered sets X and Y are said to be order-isomorphic or isomorphic or similar if there exists a one-to-one correspondence f : X → Y which preserves the order relations. Such a function f is then called an order-isomorphism or isomorphism from X onto Y.

• A) Given statement is false.
• B) Given statement is true.
• C)
• D)

An element 'a' in a well-ordered set A is called a limit element if ______________________.

• A) it has an immediate predecessor.
• B) it is the first element.
• C) it has an immediate predecessor and if it is the first element.
• D) it does not has an immediate predecessor and if it is not the first element.

Let S be a partially ordered set, and let A be a subset of S. If A has a lower bound we say A is bounded below.

• A) Given statement is true.
• B) Given statement is false.
• C)
• D)

Let C be a non-empty set of non-empty sets. A function f : C → ∪C is called a choice function for C if f(A) ϵ A for all A ϵ C.

• A) False
• B) True
• C)
• D)

The ordinal number of empty set is __________.

• A) 1
• B) 0
• C)
• D)

Let C be the set of countries on earth, thinking of each country as a collection of cities. Then ∪C is the set of all cities on earth, and the function f that assigns to each country its capital city is a choice function for C.

• A) True
• B) False
• C)
• D)

Suppose A and B are well-ordered sets, and suppose A is similar to an initial segment of B. Then B is said to be ________than A.

• A) longer
• B) shorter
• C)
• D)

Let C be the collection of all pairs of shoes in the world. Then the function that picks the left shoe out of each pair is a choice function for C.

• A) False
• B) True
• C)
• D)

A is bounded if ____________________.

• A) A has only an upper bound.
• B) None of these.
• C) A has only a lower bound.
• D) A has an upper and lower bound.

If an upper bound of A succeeds every other upper bound of A, then it is called the supremum of A

• A) Given statement is false.
• B) Given statement is true.
• C)
• D)

The ordinal number of {1,2,3,4} is _________.

• A) 4
• B) 3
• C) 2
• D) 1

Let λ and μ be ordinal numbers and let A and B be two well-ordered sets such that λ = ord(A) and μ = ord(B). Then λ < μ if A is similar to an initial segment of B.

• A) false
• B) true
• C)
• D)

Let A be a well-ordered set. Then every subset of A is well-ordered.

• A) Given statement is false.
• B) Given statement is true.
• C)
• D)