Download Chapter 2: Sets and Whole-Number Operations and Properties

Chapter 2: Sets and Whole-Number Operations and Properties
2.2 Addition and Subtraction of Whole Numbers
2.2.1. Using Models and Sets to Define Addition
2.2.1.1. How are addition and subtraction different?
2.2.1.2. How are addition and subtraction alike?
2.2.1.3. Union of Two Sets
2.2.1.3.1. Definition of the union of two sets: The union of two sets A and B is the set
containing every element belonging to set A or set B and is written A ∪ B (read “the
union of A and B”)
2.2.1.3.2. See examples on p. 74
2.2.1.4. Intersection of Two Sets
2.2.1.4.1. Definition of the intersection of two sets: The intersection of two sets A and B is
the set containing every element belonging to both set A and set B and is written A ∩ B
(read “the intersection of A and B”)
2.2.1.4.2. Two sets are said to be disjoint or mutually exclusive if their intersection is the
empty set
2.2.1.4.3. See examples p. 75
2.2.1.5. Addition of Whole Numbers
2.2.1.5.1. Definition of the addition of whole numbers: In the addition of whole numbers, if A
and B are two disjoint sets, and n(A) = a and n(B) = b, then a + b = n(A ∪ B). In the
equation a + b = c, a and b are addends, and c is the sum
2.2.1.5.2. See example p. 77
2.2.2. Basic Properties of Addition
2.2.2.1. Closure Property of Addition
2.2.2.1.1. Guarantees that the addition of two whole numbers results in another unique whole
number
2.2.2.1.2. Another way to say this is: The whole numbers are closed under addition
2.2.2.1.3. Closed means that the number obtained after an operation on any numbers in the
set yield a number in the set
2.2.2.1.4. Unique means that a + b gives one and only one result
2.2.2.1.5. For whole numbers a and b, a + b is a unique whole number
2.2.2.2. Identity Property of Addition
2.2.2.2.1. a + 0 = 0 + a = a
2.2.2.2.2. Zero is called the additive identity element
2.2.2.2.3. There exists a unique whole number, 0, such that 0 + a = a + 0 = a for every
whole number a. Zero is the additive identity element
2.2.2.3. Commutative Property of Addition
2.2.2.3.1. a + b = b + a
2.2.2.3.2. Order is no t important in addition
2.2.2.3.3. For whole numbers a and b, a + b = b + a
2.2.2.4. Associative Property of Addition
2.2.2.4.1. (a + b) + c = a + (b + c)
2.2.2.4.2. The sum of three numbers is the same regardless of which two are added together
first
2.2.2.4.3. For whole numbers a, b, and c, (a + b) + c = a + (b + c)
2.2.2.5. Definition of Greater Than (>) and Less Than (<) for Whole Numbers
2.2.2.5.1. Given whole numbers a and b, a is greater than b, symbolized as a > b, if and only if
there is a whole number k > 0 such that a = b + k. Also, b is less than a (b < a),
whenever a > b
2.2.3. Modeling Subtraction
2.2.3.1. Taking away
2.2.3.2. Separating
2.2.3.3. Comparing
2.2.3.4. See examples p. 80-82
2.2.4. Using Addition to Define Subtraction
2.2.4.1. Subtraction as the Inverse of Addition
2.2.4.1.1. Subtraction is the inverse operation of addition
2.2.4.1.2. Definition of subtraction of whole numbers: In the subtraction of whole
numbers, a and b, a – b = c if and only if c is a unique whole number such
that c + b = a. In the equation, a – b = c, a is the minuend, b is the
subtrahend, and c is the difference
2.2.4.2. Subtraction as Finding the Missing Addend
2.2.4.2.1. Addition model: addend + addend = sum
2.2.4.2.2. Subtraction model: sum – addend = missing addend
2.2.4.2.3. Re-stated Subtraction model: addend + missing addend = sum
2.2.4.2.4. Using this model: if you know addition facts, then you know subtraction facts
Problems and Exercises p. 85
2.2.2.6. Home work: 1b, 2bc, 3, 4a, 10a, 11a, 12, 15, 17, 19, 21