Other Math Answers

Hanna has taken an annuity of €140,000 to buy a new home- a loan. The term of the loan has been agreed for 12 years and the annual interest rate on the loan is set at 1,8 %. The loan is shortened monthly.

a) how big is the annuity of the loan?

b) how much of the loan is left after three years?

c) How Much Does Hanna pay in interest in total?

Prove the following by giving examples:

1. Commutative laws

4. Identity laws

2. Associative laws

5. Universal bound law s

3. Idempotent laws

1. Construct an infinite decimal .b1b2 . . . bn . . . such that ann ≠bn for each positive n.

2. Does this contradict the above assumption that [0, 1] was countable? Explain.

3. What can you conclude from the above contradiction? Explain.

4. Is the cardinal number of [0, 1] strictly greater than ℵ0? Justify your answer.

5. Is |R| strictly greater than ℵ0? Justify your answer.

... ℵ0 is the least transfinite cardinal number. If a is any transfinite cardinal number different from ℵ0, then

ℵ0 < a.

Exercise 2.49. Prove the above claim of Cantor. (Hint: Let A be an infinite set and let a = |A|. Can you define a one-to-one function from N to A? What can you conclude about the relationship between ℵ0 and a? To define such a function you will need to make countably-many choices of some elements of A. The ability to make such choices depends on what is known as the Axiom of Choice — an important but rather controversial principle in mathematics. A detailed discussion of this axiom and its history can be found in [1] and in the references therein.)

The number ℵ0 is greater than any finite number µ:

ℵ0 > µ.

Exercise 2.48. Prove the above claim of Cantor.

Show that ℵ0 is the cardinal number of N; that is, show that |N| = ℵ0. Cantor’s next claim is that ℵ0 is greater than any finite cardinal number:

1. According to Dedekind’s definition, are N, Z, and Q infinite sets? Explain.

2. Compare Cantor’s and Dedekind’s definitions.

Prove that A is an infinite set if, and only if, A is not equivalent to any finite subset of N.

2. Describe finite cardinal numbers.

Solve the following quadrantal triangle (c=90°)

1. a = 70° 7.8' b = 52°36.7'

2. A = 118° 46.4' C = 110° 7.8'