Find the cardinal number for the given set. A={6,10,14,….,58}

Question

Find the cardinal number for the given set.   A={6,10,14,….,58}

Answer 1

M = {g ∣ g ∈ ℕ, g is odd, and ∣g∣ ≤ 10}

Answer 2

The number of distinct elements in a finite set is called its cardinal number.  For example, the set {1, 2, 3} has three distinct elements, so its cardinal number is 3.  The set {1, 2, 2, 3} has four elements but only three distinct elements (1,2,3) since 2 is repeated; so its cardinal number is also 3.  Your set is:

 

A = {6, 10, 14, ..., 58}

 

Each element is 4 more than the preceding element, so it represents an arithmetic sequence with the recursive form of a_n = a_n-1 + 4.  One way to figure it out is simply to use the recursion relation to fill in the missing elements:

 

A = {6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 52, 58}

 

If you count them, it adds up to 14 and each element is distinct (no repeats), so the cardinal number of set A = 14.

 

Filling in the missing elements can be difficult if the set is large.  Fortunately, you can figure out the number of elements in set A by using the explicit form of the arithmetic sequence:

 

a_n = 6 + 4(n-1)

 

To figure out how many elements are in the set, set a_n = 58 (the last element in the set) and solve for n, the number of elements in the set:

 

58 = 6 + 4n - 4

58 = 2 + 4n

56 = 4n

14 = n

 

Since the set is an arithmetic sequence, we know that each element differs from the other elements and is distinct.  Therefore the cardinal number of set A is 14.