Question

Find the cardinal number for the given set.

A = ( 9, 11, 13, .....31)

what is the cardinal number?

Answer 1

The cardinal number is 12

Explanation:

The cardinal number of a set represents the number of distinct elements of the set. That means for this exercise the goal is to identify the total number of elements of the set.

First step. Identifying the elements of the set.

If we look at the sequence 9, 11, 13, ... , 31, we have an arithmetic sequence
`a_n`, so we can start by finding the common difference d.

`d = a_(n+1)-a_n`

That means we can pick any 2 consecutive terms and find the subtraction of them, so we can pick for example 9 and 11, thus we get

`d= 11-9\\d=2`

Second step. Finding the number of terms of the arithmetic sequence.

Now to find the number of elements n of that sequence, we can look at the last number of the sequence on this case it is 31, and we can use the formula for the nth term, that is

`a_n=a_1 + d(n-1)`

Replacing the last term
`a_n = 31` and the initial term
`a_1=9`, will give us

`31=9+2(n-1)`

From there we can solve for n, by moving first that 9 to the other side.

`31-9=2(n-1)\\22=2(n-1)`

Then we can divide both sides by 2, and lastly add by 1.

`\cfrac{22}{2} = n-1  \\11 = n-1\\\boxed{n= 12}`

Thus since the sequence has 12 elements, we know that the cardinal number of the set is 12.