To find the union of the sets [tex]\( M \)[/tex] and [tex]\( N \)[/tex], we need to follow these steps:
### Step 1: Define the sets [tex]\( M \)[/tex] and [tex]\( N \)[/tex]
Set [tex]\( M \)[/tex]: Contains all natural numbers that are multiples of 3 between 3 and 99 (inclusive). These numbers are:
[tex]\[ M = \{3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99\} \][/tex]
Set [tex]\( N \)[/tex]: Contains all prime numbers less than 19. These numbers are:
[tex]\[ N = \{2, 3, 5, 7, 11, 13, 17\} \][/tex]
### Step 2: Calculate the union of sets [tex]\( M \)[/tex] and [tex]\( N \)[/tex]
The union of two sets [tex]\( M \)[/tex] and [tex]\( N \)[/tex], denoted as [tex]\( M \cup N \)[/tex], is the set of elements that are in either [tex]\( M \)[/tex] or [tex]\( N \)[/tex] or in both. To find the union, we combine all the distinct elements from both sets.
Here are the elements in [tex]\( M \cup N \)[/tex]:
[tex]\[ 2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99 \][/tex]
So, the union of sets [tex]\( M \)[/tex] and [tex]\( N \)[/tex] is:
[tex]\[ M \cup N = \{2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99\} \][/tex]
