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Given: a ∊ ℝ, b ∊ ℝ, c ∊ ℝ, where ℝ = {Real }, then:

Addition 1A !!" +# = # +" Commutativity 1M !!" "# = # "" Commutativity 2A !!"" +## +$ = " +"# +$# Associativity 2M !!"" "## "$ = " ""# "$# Associativity " " ! ! 3M " " = "" ="#!" # $ Inverse Property 3A !!" +"""# ="""# +" = $ Inverse Property !! " " ! ! 4A !!" +" = " +" = " Element 4M !!" "" =" "" = " 5A !!" +# = $ Closure 5M !!" "# = $ Closure ! ! 6A !!" "# = " +""## (def.) 6M !!""# = " "#$"#%&!# # ' (def.) ! !

! ! ! 7 !!" "" = " "" = " Multiplication Property of ! 0 8 !!" " #" = #" "" = #" Multiplication Property of –1 9 !!""""# = " of Opposites ! 10 !!""# +$# = "# +"$ ! 11 !!""" +## ="""# +""## Property of Opposite of a Sum ! 12 !!"""## = """## = ""#!$%&!"""#""## = "# Property of Opposites in Products ! " " " ! = "!#$%! = ( &!" ) ' 13 " " % (" " Axiom of Reciprocals ! $ ' !!# "& " " " = " 14 !!"# " # Property of the Reciprocal of a !

! 15 !!" = " Reflexive Property of 16 !!"#!" = #$!%&'(!# = " Symmetrical Property of Equality 17 !!"#!" = #!$%&!# = $'!()*%!" = $ Transitive Property of Equality !

! "#!" #$!%&'(!" $ # $ Property of Equality ! 18 !! = + = + 19 !!"#!" = #$!%&'(!" "$ = # "$ Subtraction Property of Equality 20 !!"#!" = #$!%&'(!"$ = #$ Multiplication Property of Equality ! " # ! 21 "#!" = #!$%&!$ " '(!)*+%! = Division Property of Equality !! $ $ !

For all real numbers a and b, one and only one of the following ! 22 Axiom of Comparison statements is true: a < b, a = b, a > b. Every decimal represents a real and every real 23 Axiom of Completeness number has a decimal representation. 24 Between any two real numbers there is another . Property of Density

Adapted by Cary Millsap from James R. Harkey “Properties A” (1976). Visit http://carymillsap.blogspot.com.

Properties A

Given: a ∊ ℝ, b ∊ ℝ, c ∊ ℝ, where ℝ = {Real Numbers}, then:

I.A.1 !!" +# = # +" Commutative Property of Addition I.A.2 !!" +"# +$# ="" +## +$ of Addition I.A.3 !!" +"""# ="""# +" = $ Property ! I.A.4 !!" +" = " +" = " Additive Identity Element ! I.A.5 !!" +# = $ Closure for Addition ! I.M.1 !!" "# = # "" Commutative Property of Multiplication ! 1.M.2 !!"" "## "$ = " ""# "$# Associative Property of Multiplication ! " " ! " " = "" ="#!" # $ 1.M.3 !! " " Multiplicative Inverse Property ! 1.M.4 !!" "" =" "" = " Multiplicative Identity Element 1.M.5 !!" "# = $ Closure for Multiplication ! 1.M.6 !!" "" = " "" = " Multiplication Property of Zero ! II.1 !!" = " Reflexive Property of Equality ! II.2 !!"#!" = #$!%&'(!# = " Symmetrical Property of Equality ! II.3 !!"#!" = #$!# = $$!%&'(!" = $ Transitive Property of Equality ! III !!""# +$# = "# +"$ Distributive Property ! IV !!""""# = " Axiom of Opposites ! V !!""" +## ="""# +""## Property of Opposite of a Sum ! VI !!" " #" = #" "" = #" Multiplication Property of –1 ! VII !!"""#"## = """## = ""#!$%&!"""#""## = "# Property of Opposites in Products ! " " " ! = "!#$%! = ( &!" ) ' " " % (" " ! $ ' VIII !!# "& Axiom of Reciprocals " " " = " IX !!"# " # Property of the Reciprocal of a Product ! X !!" "# = " +""## Definition of Subtraction " " " ! = "" = " " #!# # $ XI !!# # # Definition of Division ! XII !!"#!" = #$!%&'(!" +$ = # +$ Addition Property of Equality XIII !!"#!" = #$!%&'(!" "$ = # "$ Subtraction Property of Equality ! XIV !!"#!" = #$!%&'(!"$ = #$ Multiplication Property of Equality ! * # ! "#!" = #$!$ " %$!&'()! = XV !! + $ Division Property of Equality ! For all real numbers a and b, one and only one of the following statements ! XVI is true: a < b, a = b, a > b. Axiom of Comparison Every decimal represents a real number and every real number has a XVII decimal representation. Axiom of Completeness Between any two real numbers there XVIII is another real number Property of Density

Adapted by Cary Millsap from James R. Harkey “Properties A” (1976). Visit http://carymillsap.blogspot.com.