Answer: 10
===========================================================
Step-by-step explanation:
Even though your teacher doesn't want you to list the items of the set, it helps to do so.
We'll be working with these two sets
A = {b, d, f, h, j, I, n, p, r, t}
C = {d, h, I, p, t}
When we union them together, we combine the two sets together. Think of it like throwing all the letters in one bin rather than two bins.
A u C = {b, d, f, h, j, I, n, p, r, t, d, h, I, p, t }
The stuff that isn't bolded is set A, while the stuff that is bolded is set C
After we toss out the duplicates, we end up with this
A u C = {b, d, f, h, j, I, n, p, r, t}
But wait, that's just set A. Notice how everything in set C can be found in set A. This indicates set C is a subset of set A.
That's why all of the stuff in bold was tossed out (because they were duplicates of stuff already mentioned).
Once we determine what set A u C looks like, we count out the number of items in that set to determine the final answer.
There are 10 items in {b, d, f, h, j, I, n, p, r, t} which means 10 is the final answer.
----------------------------
An alternative method is to use the formula below
n(A u C) = n(A) + n(C) - n(A and C)
n(A u C) = 10 + 5 - 5
n(A u C) = 10
The notation n(A and C) counts how many items are found in both sets A and C at the same time. But as mentioned earlier, this is identical to just counting how many items are in set C. So we'll have n(C) cancel out with itself.
In short, n(A u C) = n(A) = 10