To find the number of elements in the union of two sets [tex]\( U \)[/tex] and [tex]\( A \)[/tex], we start by understanding the sets individually.
1. Set [tex]\( U \)[/tex]: This set includes all positive integers less than 20. So, [tex]\[
U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}
\][/tex]
2. Set [tex]\( A \)[/tex]: This set consists of all multiples of 4 within the same range. Therefore, [tex]\[
A = \{4, 8, 12, 16\}
\][/tex]
Next, we proceed to find the union of these two sets, which includes all distinct elements from both sets combined.
[tex]\[
U \cup A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}
\][/tex]
Notice that elements in [tex]\( A \)[/tex] are already included in the complete list of [tex]\( U \)[/tex]. Thus, the union does not add any new elements beyond those already in [tex]\( U \)[/tex].
Finally, count the number of distinct elements in the union set [tex]\( U \cup A \)[/tex]:
[tex]\[
n(U \cup A) = 19
\][/tex]
Therefore, the number of elements in the union of [tex]\( U \)[/tex] and [tex]\( A \)[/tex] is given by:
[tex]\[
\boxed{19}
\][/tex]
