Which of the following sets is countable, and which has the same cardinality as
ID: 3141088 • Letter: W
Question
Which of the following sets is countable, and which has the same cardinality as R? Informal justification is acceptable. (1) {^n squareroot 2|n elementof N}. (2) {q elementof Q|q has denominator a multiple of 3 when q is expressed in lowest terms}. (3) Q intersection [2, 3). (4) [3, 4] union [5, 6]. (5) GL_3 (Z), which is the set of invertible 3 times 3 matrices with integer entries. (6) [0, 1] times [0, 1]. (7) {9^x |x elementof R}. (8) {S subsetoforequal N|S has 7 elements}. (9) The set with elements that are the closed bounded intervals in R having rational endpoints.Explanation / Answer
(According to Chegg policy, only four subquestions will be answered. Please post the remaining in another question)
1. { n2 , n N }
Since every n2 has a pre-image n in N and that a2 = b2 => a = b, there is a bijection from N to this set.
=> The set is countable.
The set does not have the same cardinality as R as N and R do not have the same cardinality.
2. { q Q | q has denominator as multiple of 3 when q is expressed in lowest terms }
Every rational number can be written in the form a/b where a and b are integers.
Since b is fixed, the only varying part is a.
a can be any integer or in other words the set has same cardinality as Z.
Since Z is countable, the given set is also countable.
The set does not have the same cardinality as R as N and R do not have the same cardinality.
3. Q [2,3)
Q [2,3) is the set of rationals between 2 and 3 excluding 3.
Since Q is itself countable, any subset of it will be countable and so the set is countable.
The set does not have the same cardinality as R as Q and R do not have the same cardinality.
4. [3,4] U [5,6]
The set [3,4] is uncountable as it is a subset of R and (3,4) is uncountable.
Similarly [5,6] is uncountable
Thus [3,4] U [5,6] is uncountable.
We know that two non-empty open intervals have same cardinality. Thus (3,4) and (5,6) have same cardinality as R. By extension, [3,4] and [5,6] have same cardinality as R.
Thus [3,4] U [5,6] has same cardinality as R.
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