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Level 158

Field Properties & Axioms of Equality


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Commutative of addition
add in any order
Commutative of multiplication
multiply in any order
Associative of addition
add by regrouping
Associative of multiplication
multiply by regrouping
identity of addition
sum of a number and zero=the number
Identity multiplication
product of a number and 1 equals the number
Inverse property of zero for addition
adding a number to its opposite equals zero
Inverse Property of Multiplication
The product of a nonzero number and its multiplicative inverse, or reciprocal, is 1.
Multiplication property of zero
a number multipied by zero=zero
Property of opposites
a number multiplied by -1 is the opposite of the number
Addition Property of Equality
For all real numbers x, y, and z, if x = y, then x + z = y + z.
Multiplication Property of Equality
For all real numbers x, y, and z, if x = y, then xz = yz.
Substitution Property of Equality
For all real numbers x and y, if x = y , then y can be substituted for x in any expression and vice versa.
a=a
Reflexive Property of Equality
Symmetric Property of Equality
For all real numbers x and y, if x = y, then y = x.
Transitive Property of Equality
For all real numbers x, y, and z , if x = y and y = z, then x = z.
a+b=b+a
Commutative Property of Addition
ab=ba
Commutative Property of Multiplication
(a+b)+c=a+(b+c)
Associative Property of Addition
(AB)C = A(BC)
Associative Property of Multiplication
a+0=a
Identity Property of Addition
1a=a
Identity Property of Multiplication
a+(-a)=0
Inverse Property of Addition
a(1/a)=1
Inverse Property of Multiplication
a(b+c)=ab+ac
Distributive Property
Closure Property of Addition
If a and b are real numbers, then the sum of a+b is a real number
Closure Property of Multiplication
If a and b are real numbers, then the product of "a" times b is a real number
x-y=x+(-y)
Definition of Subtraction
x÷y=x(1÷y)
Definition of division
0(x)=0
multiplicative property of 0
-1(x)=-x
multiplicative property of -1
1/xy=1/x(1/y)
reciprocal of a product
multiplication prop of fractions
ab/xy= a/x(b/y) x≠0, y≠0
Closure property- A set is closed under an operation if and only if...
the operation on two elements of the set produces another element of the set. If an element outside the set is produced, then the operation is not closed.
Trichotomy
For any a and b, exactly one of the following is true:
Converse APE
If a+c=b+c, then a=b
Prop of opp of sum
For all real number a and b:
Cancellation property of additive inverses
for all real numbers a, -(-a)=a
(-a)b=-ab
Property of opp in products
Reflexive Property of Equality
Any number equals itself. x = x
Transitive Property of Equaity
*the property that states if a = b and b = c, then a = c