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Chapter 1

Sets and Numbers

 
1.3 Operations and Properties in the Set R

With any two rational numbers a,b we can define in a unique way
 
  • their sum a+b defining addition
  • their product a b defining multiplication 
These two operations have some basic properties.

(P1) If a,b,c are any three real numbers then:
 
            a + ( b + c ) = ( a + b ) + c        Associative Law in Addition

(P2) If a is any real number then
 
            a + 0 = 0 + a = a

(P3) For every real number a, there is a number -a such that

            ( -a ) + a = a + ( -a ) = 0

(P4) If a,b are any real numbers, then

            a + b = b + a                            Commutative Law in Addition

(P5) If a,b,c are real numbers then

            ( b  c ) = ( a  b )  c   Associative Law in Multiplication      

(P6) If a is any real number then

            1 = 1   a  = a

(P7) For every number there is a number  such that

 

(P8) If a,b are any real numbers, then

            b = b  a                          Commutative Law in Multiplication

(P9) For any numbers a,b,c that belong to the set of real numbers

            ( b  + c)  = a  b + a Distributive Law

The subtraction and the division are defined with the help of addition and multiplication respectively, as below:
 
  • a - b = a + ( -b)
For the four operations the following rules apply:

(P10) If a=b and c=d then

              a + c  = b + d  and a  c = b  d

        That is we can add and multiply the sides of two equations.

(P11) If a=b then

              a + c = b + c  and a  c = b  c

          That is, we can add the same number to both of the sides of an equation.
          Also, we can multiply its side of an equation with the same number.
 
(P12)  If a + c = b + c then a = b
            If a  c = b  c and c is not equal to 0, then a = b.

            So, we can delete from both the sides of an equation  a common addendum or the
            same non zero factor.

(P13) 0 = 0
           If a  b = 0 then a = 0 or b = 0

(P14) ( -1 )  a  = - a
           ( -a )  b = -  b
           ( -a )  ( - b ) = a  b

(P15) - ( a+b ) = - a - b
 

(P16)