To solve the problem of finding the number of elements in the set ( {x \mid x \text{ is a positive integer, } x \leq 1000, \text{ and } x \text{ is not divisible by 3 or 5} ), we use logical equivalence and basic counting:

Step 1: Understand the logical condition

The condition "x is not divisible by 3 or not divisible by 5" is equivalent to not (x is divisible by both 3 and 5). This is because the only numbers where both "divisible by 3" and "divisible by 5" are true are those divisible by their least common multiple (LCM of 3 and 5 = 15).

Step 2: Count numbers divisible by both 3 and 5

Numbers divisible by 15 up to 1000:
( \left\lfloor \frac{1000}{15} \right\rfloor = 66 ) (since ( 15 \times 66 = 990 ), and ( 15 \times 67 = 1005 > 1000 )).

Step 3: Calculate the desired count

Total positive integers ≤1000: 1000.
Subtract the count of numbers divisible by both 3 and 5:
( 1000 - 66 = 934 ).

Answer: (\boxed{934})

作者声明:本文包含人工智能生成内容。