MTH104 Final Term Quiz 3 6 July 2023 by Miss Mehwish

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Q1. 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).

1. True
2. False

Correct option: a) True

Q2. 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.

1. True
2. False

Correct option: b) False

Q3. The ordinal number of set {a, {b, c}} is ________________.

1. 0
2. 1
3. 2
4. 3

Correct option: d) 3

Q4. A is bounded if ____________________.

1. A has only a lower bound.
2. None of these.
3. A has only an upper bound.
4. A has an upper and lower bound.

Correct option: d) A has an upper and lower bound.

Q5. The ordinal number of the set P of counting numbers is denoted by ω.

1. True
2. False

Correct option: a) True

Q6. 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.

1. Shorter
2. Longer

Correct option: a) Shorter

Q7. If an upper bound of A succeeds every other upper bound of A, then it is called the supremum of A. Options:

1. False
2. True

Correct option: b) True

Q8. Let S be a partially ordered set, and let A be a subset of S. An element n in S is called a lower bound of A if n precedes every element of A, that is, for every y ϵ A, we have y ≥ n.

1. False
2. True

Correct option: b) True

Q9. 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.

1. Longer
2. Shorter

Correct option: a) Longer

Q10. 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.

1. True
2. False

Correct option: b) False