Mathematical Systems The Real Numbers Mathematical Systems . The real numbers, R, is the set of all . A mathematical system is a set with one or unending decimals. more binary operations defined on it. Mathematical Systems – If the unending decimal repeats the number – A binary operation is a rule that assigns to 2 belongs to Q and is rational. elements of a set a unique third element. 0.5 = 0.5000… = 1/2 . If 5 and 7 belong to N and addition is the binary 0.33333… = 1/3 Groups operation then 12 Is the unique “answer.” – If the unending decimal does not repeat the 5 + 7 = 12 number belongs to the set of irrational numbers. If 4 and 4 belong to I and subtraction is the binary 1.41421…≈ 2. operation then 0 is the unique “answer.” Symbols are used for irrational numbers. 4 – 4 = 0 Chapter 5 – Section 5 and Chapter 10 In-class Assignment 22 -1, 2 No in-class assignment problem Properties of Real Numbers Properties of Real Numbers Properties of Real Numbers Closure Associative Property Commutative Property . If an operation is performed on any two members . Given 3 numbers of the set in the same order and an . Given 2 numbers of a set and an operation operation the result is the same regardless of the of the set and the result is a member of the set the result is the same regardless of the then the set is closed. grouping. (2 + 5) + 4 = 2 + (5 + 4) – If the set is W and the operation is subtraction then 7 − order of the numbers. 12 does not belong to W. W is not closed under – Notice – order is the same just the grouping is 5 + 6 = 6 + 5 but 4 − 2 ≠ 2 − 4 subtraction. different. 7 + 4 = 2 + 9 – If the set is I and the operation is subtraction then 7 − 12 (16 8) 2 ≠ 16 (8 2) = −5. This implies I might be closed under subtraction. – Generally the set R has the commutative . The set R is closed under addition and Generally the set R has the associative property under property under addition and multiplication, but multiplication. It is not closed under subtraction addition and multiplication but not under subtraction not under subtraction and division. and division. and division. In-class Assignment 22 - 3 In-class Assignment 22 - 4 In-class Assignment 22 - 5 1 Mathematical Systems Properties of Real Numbers Properties of Real Numbers Determining Which Properties an Identity Property Inverse Property Infinite Systems Has - Closure . A set has the identity property if contains a unique . E is {2, 4, 6, 8, …} and 6 + 24 = 30, element, a, such that the element in operation with . A set under an operation, , has the inverse any other element in the set in any order results in property if for each element, a, of the set addition 30 E that number. there is another element, a-1, (called a . Closure – . Symbolically: a + b = b + a = b -1 -1 0 + 8 = 8 + 0 =8 and 1 x 15 = 15 x 1 = 15 inverse) such that a a = a a = identity. – choose 2 numbers 246 + 12 = 258, Suggests 0 is the identity for addition and 7 + −7 = −7 + 7 = 0, which is the additive identity – Add 258 E 1 is the identity for multiplication. 23 7 − 0 ≠ 0 − 7 and 9 1 ≠ 1 9 1 , which is the multiplicative identity, – Does the answer 32 belong to E? E may be closed . There is no identity element for subtraction or division. – Repeat several times. The identity property allows the operation to be . The inverse property allows for the under addition. performed without anything happening. operation to be undone. In-class assignment problem 22 - 6 In-class Assignment 22 - 7 In-class Assignment 22 - 8 Determining Which Properties an Determining Which Properties an Determining Which Properties an Infinite Systems Has - Associative Infinite Systems Has – Commutative Infinite Systems Has - Identity . E is {2, 4, 6, 8, …} and 2+(12+8) = 2+20 = 22 . E is {2, 4, 6, 8, …} and . 16+24 =40 . E is {2, 4, 6, 8, …} and addition (2+12)+8 = 14+8 = 22 addition . 24+16 =40 addition 10 + ? = 10 – Choose any number in E. – Choose 3 elements of E. – Choose any 2 elements of E. – Try to find another 344 + ? = 344 – Keep same order but (6+4)+8 = 10+8 = 18 – Add . 8+14 = 22 number to add to the chosen number so that different grouping. 6+(4+8) = 6+14 = 18 – Change the order of the . 14+8 = 22 Only 0 would work – the answer is the chosen. so no identity. – Are the answers the same? elements and add – An identity must work for – Repeat several times. – Are the answers the same? all numbers of the set. E may have the . E may have E does not have the associative – Repeat several times. the commu- – If no identity then the set identity property property under tative property can have no inverse under addition property. addition. under addition. and no inverse property. In-class Assignment 22 - 9 In-class Assignment 22 - 10 In-class Assignment 22 - 11 2.