Determine whether each of these sets is finite, countably infinite, or uncountable. for those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.

a) the integers greater than 10
b) the odd negative integers
c) the integers with absolute value less than 1,000,000
d) the real numbers between 0 and 2
e) the set a×z+ where a={2,3}

a) Countably infinite

b) Countably infinite

c) Finite

d) Uncountable

e) Countably infinite

Step-by-step explanation:

a) Let S the set of integers grater than 10.

Consider the following correspondence:

defined by for .

Let's see that the function is one-to-one.

Suppose that f(10+k)=f(10+j) for k≠j. Then k-1=j-1. Thus k-j=1-1=0. Then k=j. This implies that 10+k=10+j. Then the correspondence is injective.

b) Let S the set of odd negative integers

Consider the following correspondence:

defined by .

Let's see that the function is one-to-one.

Suppose that f(-(2k+1))=f(-(2j+1)) for k≠j. By definition, k=j. This implies that the correspondence is injective.

c) The integers with absolute value less than 1,000,000 are in the intervals A=(-1.000.000, 0) B=[0, 1.000.000). Then there is 998.000 integers in A that satisfies the condition and 999.000 integers in B that satifies the condition.

d) The set of real number between 0 and 2 is the interval (0,2) and you can prove that the interval (0,2) is equipotent to the reals. Then the set is uncountable.

e) Let S the set A×Z+ where A={2,3}

Consider the following correspondence:

defined by

Let's see that the function is one-to-one.

Consider three cases:

1. , then 2k=2j, thus k=j.

2. , then 2k+1=2j+1, then 2k=2j, thus k=j.

3.  , then 2k=2j+1. But this is impossible because 2k is an even number and 2j+1 is an odd number.

Then we conclude that the correspondence is one-to-one.

answer:

it is general formula that is used to write value to its excet position

a11 a12 a13

a21 a22 a23

it is the example of order 2 by 3 matrix in this

a11 means value occur in 1 st row 1st column

a12 means value occurs in 1st row 2nd column

a13 means value occur in 1st row and 3rd column

a21 means value occurs in 2nd row and 1st column

a22 means value occurs in 2nd row and in 2nd column

a23 means value occurs in 2nd row and in 3rd column

answer: the angel abc is 30°,and the abc is a triangle where the angel bac is 90°

so ac is the half of 24=12ft,and from the pythagorean theorem we find ba

ba=√bc^2-ac^2=√576-144=√432=12√3ft

so bac is 12+12√3=12(1+√3)ft