Language learning | English » Sets of numbers, cardinality of sets

Sets of numbers, cardinality of sets Take the set of natural numbers as a starting point. The set of natural numbers consists of non-negative whole numbers. Its sign is N (If we only want to refer to the positive whole numbers we sign it with N+ . Consider two basic operations in the set of natural numbers: addition and multiplication If a and b are elements of N, then a+b and ab are also elements of N so these operations do not lead out from the set. If an operation in the set does not lead out of the set( i.e the result of the operation will also be element of the set), then the set is closed on this operation. If the result of an operation in the set is not the element of the set then the set is open on that operation. So the set of natural numbers is closed on addition and multiplication • Most important properties of these operations: a+b=b+a , ab=ba - commutative property (parts are exchangable) (a+b)+c=a+(b+c) , ab(c)=(ab)c - associative property (a+b)c=ac+bc - distributive

property (addition to multiplication only) We also accept the division with remainder in the set of natural numbers. If a>b and c is element of N, then there exsits a natural number m<b for which a=cb+m is true. The remainder of the division is m (it can also be 0). If m=0, then a is divisible by b Next we introduce the prime numbers • Def.: We call p positive integer a prime number if and only if it has two divisors, 1 and itself. If a natural number greater than 0 is not a prime than it is called a complex number (there are infinitely many prime and complex numbers) Theorem: every complex number can be written up as the product of prime numbers • With the help of this theorem we can define the greatest common divisor, smallest common multiple of two or more positive integers, or the number of divisors of a positive integer etc. • If two positive integers have no common divisors out of 0, then they are called relative primes • Most important rules of divisibility o With

2 – only even numbers o With 3 – if the sum of the digits is divisible by 3 o With 4 – if the number formed by the last two digits is divisible by 4 o With 5 – if the numbers ends in 5 or 0 o With 6 – if it is divisible by 3 and 2 o With 8 – if the number formed by the last 3 digits is divisible by 8 o With 9 – if the sum of the digits is divisible by 9 • Other important rules of divisibility: (let a,b,c and n be elements of the set of positive natural numbers o o o o If n is a divisor of a and b as well then n is a divisor af a+b and a-b If n is a divisor of a, and a is a divisor of b then n is a divisor of b If n is a divisor of ab , and (n; a)=1, then n is a divisor of b If a is a divisor of n, and b is also a divisor of n, and (a; b)=1, then ab is a divisor of n Investigating the operation of subtraction, we get that if a and b are elements of N, then a-b is not necessarily an element of N. Therefore the set of natural numbers is an open set on subtraction,

and this way leads us to the set of negative numbers. The negative and positive integers and 0 are together called whole numbers, the sign of the set is Z. Set Z is closed on addition, multiplication and subtraction as well, and the properties of the set of natural numbers are true in this set as well (commutative, distributive, associative) On division, Z is open, because if a and b are elements of the set of whole numbers, then a/b will not necessarily be an integer these numbers will be fractions Def.: if a and b are integers and b is not 0, the we call a/b rational number So rational numbers are numbers that can be written up as the ratio of two integers. The sign of the set of rational numbers is Q, and this set is closed on all four basic operations, since ab, a/b, a+b and a-b are all rational numbers if a and b are. (0 * X = a equation has no solution if a is not 0, because 0 X is 0 for all X. On the other hand if a =0 then all the rational numbers would be solutions of the

equation, therefore we do not allow divising by 0 in the set of rational numbers. ) There are operations which lead us out of the set of rational numbers: for example taking the sqare root of a rational number may give us a number which can not be written up as the ratio of two integers. Def.: Those numbers which can not be written up as the ratio of two inetegers are called irrational numbers ( the set of irrational numbers is marked by Q*.) Theorem: square root of 2 is irrational Proof: indirect proof • Let’s suppose that the sqare root of 2 can be written up as the ratio of two inetegers p/q where p and q are relative primes • This way we can get: 2q2=p2 that means that p2 is even therefore p is even, so p=2k • Then: 2q2=(2k)2=4k2 therefore q2 = 2k2 • This last eqation means that q2 therefore q is even. So if the square root of 2 could be written up as the ratio of two relative primes, then both number would be even. But they can not be since they are relative primes,

therefore our statement was false Rational numbers – véges vagy végtelen szakaszos tizedes törtek Irrational numbers – végtelen, nem szakaszos tizedes törtek Q and Q* are disjunct sets because ther intersection is the empty set. Def.: the union of Q and Q* is the set of real numbers R • There is a one to one function between the elements of R and the numbers on the numberline, therefore we can demonstrate them on the numberline We define the number of elements in a set with the cardinality of the set • If set A has finitely many elements (n elements) then its cardinality is n Def.: if a one to one function is between the elements of two sets, then the cardinality of the two sets are equal • E.g: positive integers N+ and positive even integers o For every element n we can find 2n in the other set and the other way around, so there is a one to one function between the elements of the two sets Def.: If the cardinality of any set H is equal to that of N+ then H has a

countable cardinality • E.g Q has the same cardinality as N+ • E.g R has an uncountable cardinality Applications: basic arithmetical problems, calculation with fractions, introduction of units, construction of sqare root n