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Mathematical Systems

The Real Mathematical Systems

. The real numbers, R, is the 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 – A binary is a rule that assigns to 2 belongs to Q and is rational. elements of a set a unique third . 0.5 = 0.5000… = 1/2 . If 5 and 7 belong to N and 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 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- Assignment 22 -1, 2 No in-class assignment problem

Properties of Real Numbers Properties of Real Numbers Properties of Real Numbers . 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 , but multiplication. It is not closed under subtraction addition and multiplication but not under subtraction not under subtraction and . 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 . 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 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

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