Question

5. For each set below, say whether it is finite, countably infinite, or uncountable. Justify your answer in each case, giving a brief reason rather than an actual proof.

a. The points along the circumference of a unit circle.

(Uncountable because across the unit circle because points are one-to-one correspondence to real numbers) so they are uncountable

b. The carbon atoms in a single page of the textbook.

("Finite", since we are able to count the number of atoms in a single page of textbook)(The single page is the limit and it contains a number of carbon atom elements)

c. The different angles that could be formed when two lines intersect (e.g. 30 degrees, 45 degrees, 359.89 degrees, etc….)

("uncountable" because, the different angles that can be made would be in radial from 0-2pi, pi is an example of irrational angle and cannot be counted in the set.

d. All irrationals which are exact square roots of a natural number.

("countably infinite", there are infinite perfect squares as x approaches infinity, so there are infinite exact square roots for a natural number, which is one to one correspondence and is onto therefore is countable

e. All irrationals of the form a+sqrt(b) where a and b are rational numbers.

(im unsure about this one but i would say "uncountable" because not all a+sqrt(b) will be rational, sqrt(b) would have to be ration for it to be a countable, since irrational numbers are countable

f. The set of all squares that can be drawn within a unit circle.

"Uncountable" it has a one to one correspondence from 0 to 2pi

i wanted to crosscheck n see if this is right

Answer 1

Part (a) and (b) are correct, along with the reasons mentioned.

For part (c), the answer is "uncountable".(correct).

The reasoning for (c) has a minor error. Although the set has an angle whose value is irrational, it cannot be concluded that the set is uncountable. For example, the set is a countably infinite set.

Correct reason: The set of such angles is equal to the closed interval , which is uncountable. (In real number line, any closed interval of the form , where is uncountable)

Part (d) is correct, along with the reason.

For part (e), the answer is countable.

Recall that the rational numbers are countable.

The set of all numbers of the form has one-to-one correspondence with the ordered pair (a,b)

Since the finite Cartesian Product of countable sets is countable, it can be concluded that the required set is countable.

(f) The required set is uncountable.

See that the length of diagonal of any square on the above circle is between to .

So, the number of such squares corresponds to the interval , which is uncountable.

P.S.: Please upvote if you have found this answer helpful.