What is the cardinal number of {1,2,4,8,16,…,1024} and how do you find the number? Is there a formula for finding it?
The Correct Answer and Explanation is :
The set {1,2,4,8,16,…,1024}\{1, 2, 4, 8, 16, \dots, 1024\} consists of powers of 2. The first element is 20=12^0 = 1, the second is 21=22^1 = 2, the third is 22=42^2 = 4, and so on. The last element in this sequence is 210=10242^{10} = 1024. This indicates that the set contains all the powers of 2 from 202^0 to 2102^{10}.
Cardinal Number:
The cardinal number of a set is the number of elements it contains. To find the cardinal number of this set, we need to determine how many terms are included from 202^0 to 2102^{10}. These terms form an arithmetic progression where the first term is 202^0, and the common ratio is 2.
There are 11 terms in total: 20,21,22,…,2102^0, 2^1, 2^2, \dots, 2^{10}. Therefore, the cardinal number of the set is 11.
Formula for Finding Cardinality:
In general, for a sequence of powers of a number bb, from b0b^0 to bnb^n, the number of terms (cardinal number) can be found by the formula: Cardinality=n+1\text{Cardinality} = n + 1
where nn is the exponent of the highest power in the sequence.
In our case, the sequence is 20,21,22,…,2102^0, 2^1, 2^2, \dots, 2^{10}, so the highest exponent is 10. Using the formula: Cardinality=10+1=11\text{Cardinality} = 10 + 1 = 11
Thus, the cardinal number of the set {1,2,4,8,16,…,1024}\{1, 2, 4, 8, 16, \dots, 1024\} is 11.
Explanation:
This set consists of numbers that are successive powers of 2. The first element is 11, which is 202^0, and the last element is 10241024, which is 2102^{10}. The sequence includes all integers from 202^0 to 2102^{10}, and since these are powers of 2, they are distinct. Therefore, the total number of elements is equal to the number of terms in the sequence, which is 11.