This paper presents a new method for generating all prime numbers up to a particular number m∈N,m⩾9, by using the set theory. The proposed method is explicit and works oriented in finding the prime numbers in order. Also, we give an efficiently computable explicit formula which exactly determines the number of primes up to a particular number m∈N,m⩾9. For the best of our knowledge, this is the first exact formula given in literature. For the sake of comparison, a unified framework is done not only for obtaining explicit formulas for the well-known sieves of Eratosthenes and Sundaram but also for obtaining exact closed form expression for the number of generated primes using these two sieve methods up to a particular number m∈N,m⩾9. In addition, the execution times are calculated for the three methods and indicate that our proposed method gives a superior performance in generating the primes.