The set of prime numbers that are less than 15 is {2, 3, 5, 7, 11, 13}.
In order to understand why these numbers are considered prime, we need to define what a prime number is. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In simpler terms, a prime number has exactly two distinct positive divisors: 1 and itself.
Now, let’s look at the numbers less than 15:
- 2: Divisors are 1 and 2 (prime)
- 3: Divisors are 1 and 3 (prime)
- 4: Divisors are 1, 2, and 4 (not prime)
- 5: Divisors are 1 and 5 (prime)
- 6: Divisors are 1, 2, 3, and 6 (not prime)
- 7: Divisors are 1 and 7 (prime)
- 8: Divisors are 1, 2, 4, and 8 (not prime)
- 9: Divisors are 1, 3, and 9 (not prime)
- 10: Divisors are 1, 2, 5, and 10 (not prime)
- 11: Divisors are 1 and 11 (prime)
- 12: Divisors are 1, 2, 3, 4, 6, and 12 (not prime)
- 13: Divisors are 1 and 13 (prime)
- 14: Divisors are 1, 2, 7, and 14 (not prime)
Thus, the only numbers that meet the criteria of having exactly two distinct positive divisors are 2, 3, 5, 7, 11, and 13. Therefore, the prime numbers less than 15 are {2, 3, 5, 7, 11, 13}.