Sect 1.3 Axioms for the Real Numbers
Axioms
are basic statements that are assumed to be true. Binary operations
(addition and multiplication) assigns any two real numbers to a third
real number.
Axioms:
- Closure
for addition when you add two real numbers, you get a real number.
- Associative
for addition it does not matter what order you associate and add
numbers.
- Identity
for addition 0 is the additive identity; 0 added to any number gives the
number itself.
- Commutative
for addition it does not matter if the order of addition is changed.
- Additive
inverse an element when added to any real number gives 0.
- Closure
for multiplication any two real numbers multiplied together gives a real
number.
- Associative
for multiplication when real numbers are multiplied, they can be
associated in any way.
- Commutative
for multiplication it does not matter if the order of multiplication is
changed.
- Identity
for multiplication 1 is the multiplicative identity; a real number
multiplied by 1 gives the number itself.
- Multiplicative
inverse (Reciprocal) an element when multiplied by any real number gives
1.
Substitution Principle:
Changing the numeral by which a number is named in an
expression involving sums or products does not change the value of the expression.
Ex. For what operation is the set
{-1,0,1} closed?
(Closed in
the case means that if we perform the operations of addition or multiplication
on the numbers in the set, will we get a result that is in the set.)
Addition: 1 + 1 = 2 Since 2 is not in the set, it is NOT
closed under addition.
*To show not closed you need
provide only one example.
Multiplication: (-1)(-1) = 1
(-1)(0) = 0
(1)(1) = 1
(1)(0) = 0
Since all
the results are in the set, it IS closed under multiplication.