7. Axioms+ 8.Applications Flashcards
Axioms
Statements about mathematics that are accepted without proof
Theorems
Statements about mathematics requiring proof; starts from axioms and follows a logical path to the theorem you wanna prove.
Closure axiom?
Given a set and an operation, we say that a set has closure for that operation if we can guarantee that the result of performing that operation on any elements in the set will result in another element of that same set.
Closure axiom for wholes?
Addition and multiplication are closed. Subtraction and division are not
Closure axiom for integers?
Addition, multiplication, and subtraction are closed. Division is not.
Closure axiom for reals?
Addition, multiplication, subtraction, and division are all closed.
a=a
Reflexive property
If a=b then b=a
Symmetric property
If a=b and b=c, then a=c
Transitive property
Commutative property of addition?
a+b=b+a
Associative property of addition?
a+(b+c)=(a+b)+c
Identity property of addition?
a+o=a
Commutative property of multiplication?
a(b)=b(a)
Associative property of multiplication?
A(bc)=(ab)c
Identity property of multiplication?
a*1=a
Multiplicative inverse?
a*1\a=1;a not equal to 0
Distributive property
a(b+c)=(ab)+(ac)
Zero property?
a*o=o
General properties?
Reflexive
Symmetric
Transitive
Properties of addition?
Commutative
Associative
Identity
Additive inverse
Properties of multiplication?
Commutative Associative Identity Multiplicative inverse Distributive Zero
Additive inverse property?
a+(-a)=0
To find the value of a multi-step arithmetic expression?
A definite order of operations must be followed
Order of operations?
Parenthesis Exponents Multiplication Division from left to right Addition Subtraction from left to right
