Determine whether each of the following sets is countable or uncountable. Justif

Question

Determine whether each of the following sets is countable or uncountable. Justify your answer.

a) The set of all straight line in the Cartesian plane, each of which passes through the origin and a point having both coordinates rational. [Note that if this set is countable, then an uncountable set of lines y = mx manages to miss all the points (p,q) with rational coordinates, except for (0,0).

b) The set of all intervals on the real line having both endpoints rational.

c) Any infinite set of non-overlapping intervals on the real line.

Explanation / Answer

Solution:

(a) Countable

Explanation:

If a line goes through co-oditnates (R1, R2) with R1 and R2 rational, then its intersection with the like x=1 is also rational, so (again excluding x=0) the lines of this type are in 1-1 correspondence with the rational points of x=1, so are countable
(b)

Countable

explanation:

the intervals are in 1-1 correspondence with a subset of QxQ, so are countable

(c)

UnCountable

Explaination:

assuming the intervals can't be single points, each contains a rational so the set is countable. If single-point intervals are allowed it can be uncountable - e.g. the set of [x,x] for all x in R.

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