Q1. set of natural numbers from 4 to 12 in builder set form Q2. set of first six positive prime number if u help it would be much appreciated plz
Answers
Answer:
1.
The standard sets of numbers can be expressed in all the three forms of representation of a set i.e., statement form, roster form, set builder form.
1. N = Natural numbers
= Set of all numbers starting from 1 → Statement form
= Set of all numbers 1, 2, 3, ………..
= {1, 2, 3, …….} → Roster form
= {x :x is a counting number starting from 1} → Set builder form
Therefore, the set of natural numbers is denoted by N i.e., N = {1, 2, 3, …….}
2. W = Whole numbers
= Set containing zero and all natural numbers → Statement form
= {0, 1, 2, 3, …….} → Roster form
= {x :x is a zero and all natural numbers} → Set builder form
Therefore, the set of whole numbers is denoted by W i.e., W = {0, 1, 2, .......}
3. Z or I = Integers
= Set containing negative of natural numbers, zero and the natural numbers → Statement form
= {………, -3, -2, -1, 0, 1, 2, 3, …….} → Roster form
= {x :x is a containing negative of natural numbers, zero and the natural numbers} → Set builder form
Therefore, the set of integers is denoted by I or Z i.e., I = {...., -2, -1, 0, 1, 2, ….}
4. E = Even natural numbers.
= Set of natural numbers, which are divisible by 2 → Statement form
= {2, 4, 6, 8, ……….} → Roster form
= {x :x is a natural number, which are divisible by 2} → Set builder form
Therefore, the set of even natural numbers is denoted by E i.e., E = {2, 4, 6, 8,.......}
5. O = Odd natural numbers.
= Set of natural numbers, which are not divisible by 2 → Statement form
= {1, 3, 5, 7, 9, ……….} → Roster form
= {x :x is a natural number, which are not divisible by 2} → Set builder form
Therefore, the set of odd natural numbers is denoted by O i.e., O = {1, 3, 5, 7, 9,.......}
Therefore, almost every standard sets of numbers can be expressed in all the three methods as discussed above.
2.A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
Groups of two to twelve dots, showing that the composite numbers of dots (4, 6, 8, 9, 10, and 12) can be arranged into rectangles but prime numbers cannot
Composite numbers can be arranged into rectangles but prime numbers cannot
The property of being prime is called primality. A simple but slow method of checking the primality of a given number {\displaystyle n}n, called trial division, tests whether {\displaystyle n}n is a multiple of any integer between 2 and {\displaystyle {\sqrt {n}}}{\sqrt {n}}. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.
Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.
Definition and examples
Step-by-step explanation: