Please help with the following. Whoever provides the most answers will be awarde

Question

Please help with the following. Whoever provides the most answers will be awarded the points. Thank you.

For each of the following, give an example of a function f:N rightarrow N and provide examples of why the function meets the requirement to be: One-to-one but not onto Onto but not one-to-one Both one-to-one and onto (cannot be the identify function) Is neither one-to-one nor onto [6|40.24] Is the union of two countable sets also countable? If so, why? If not, why not? [3|D.23] What is required for a function to have an inverse? [8|19.38] Given sets A, B, and C, clearly show that (A - B) - C = (A - C) - (B - C ). [8|19.38] Given two matrices A and B of n times n with AB = BA = In, then B is known as the inverse of A, and is unique, and A is said to be invertible. The notation B = A-1 denotes this fact. If A is the 2 times 2 matrix and if ab - bc = 0, then show for all cases that the matrix C is an inverse of A, that is, equal to A-1, if

Explanation / Answer

1.

(a)

f(n) = 2n

(b)
f(n) = n-1 for n>=2, f(1) = 1

(c)

f(2k) = 2k+1,

f(2k+1) = 2k

(d)

f(n) = 1

2.

Yes

3.

f has to be both one-one and onto

4.

let x belongs to (A-B)-C

<=>

x belongs to A-B and x doesnt belong to C

<=>

x belongs to A and x doesnt belong to B and x doesnt belong to C

<=>

(x belongs to A and doesnt belong to C), (x doent belong to B)

<=>

x belongs to A-C, and doesnt belong to B-C

<=>

x belongs to (A-C) - (B-C)

=>

(A-B)-C = (A-C)-(B-C)

thus proved

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