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Sets and Classification of Numbers Section R.2
Set Notation <ul><li>A  set  is a collection of well-defined objects. </li></ul>C  = {2, 4, 6, 8, 10} This method of repre...
Set-Builder Notation <ul><li>Another way to denote a set is to use  set-builder notation . </li></ul>C  = { x  |  x  is an...
Set Notation <ul><li>Set Notation </li></ul><ul><li>If two sets  A  and  B  have the same elements, then we say that  A  e...
Venn Diagrams <ul><li>Venn diagrams  are pictures that help us to visualize relations among sets. </li></ul>U  = {0, 2, 4,...
Using Set Notation <ul><li>Example : </li></ul><ul><li>Let  A  = {2, 4, 6, 8, 10},  B  = {2, 4, 6} and  C  = {2}.  Write T...
Classification of Numbers <ul><li>The  natural numbers , or  counting numbers , are the numbers in the set  </li></ul>The ...
Classification of Numbers <ul><li>The  integers  are the numbers in the set  </li></ul>The set of rational integers are th...
Rational Numbers <ul><li>Examples of rational numbers are </li></ul>Rational numbers can also be represented as decimals. ...
Irrational Numbers <ul><li>An  irrational   number  is a decimal that neither terminates or repeats. </li></ul>These decim...
The Real Numbers Irrational numbers Rational numbers Integers Whole Numbers Natural Numbers
Classifying Numbers <ul><li>Example : </li></ul><ul><li>List the numbers in the set </li></ul><ul><li>that are </li></ul>(...
Truncating Decimals <ul><li>To  truncate  a decimal, drop all the digits that follow the specified final digit in the deci...
Rounding Decimals <ul><li>To  round  a decimal, identify the final digit in the decimal.  If the next digit is 5 or more, ...
The Real Number Line <ul><li>The real numbers can be represented by points on a line called the  real number line .  </li>...
Plotting Points Example : On the real number line, label the points with the coordinates 0, 1.75, – 2.5, 4. Real numbers a...
Ordering Numbers Example : Replace the ? With < , > or =. (a)  3  ?  –2 Since 3 lies to the right of  –2, we know that 3 >...
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  1. 1. Sets and Classification of Numbers Section R.2
  2. 2. Set Notation <ul><li>A set is a collection of well-defined objects. </li></ul>C = {2, 4, 6, 8, 10} This method of representing a set is called the roster method . C represents the set. Braces are used to enclose the elements .
  3. 3. Set-Builder Notation <ul><li>Another way to denote a set is to use set-builder notation . </li></ul>C = { x | x is an even digit} When a letter can be used to represent any digit, it is called a variable . Letters are used to represent numbers. The bar is read “such that.”
  4. 4. Set Notation <ul><li>Set Notation </li></ul><ul><li>If two sets A and B have the same elements, then we say that A equals B and write A = B . </li></ul><ul><li>If every element of a set A is also an element of set B , then we say that A is a subset of B and write A  B . </li></ul><ul><li>If A  B and A  B , then we say that A is a proper subset of B and write A  B . </li></ul><ul><li>If a set A has no elements, it is called the empty set , or null set , and is denoted by the symbol  or { }. The empty set is a subset of every set. </li></ul>
  5. 5. Venn Diagrams <ul><li>Venn diagrams are pictures that help us to visualize relations among sets. </li></ul>U = {0, 2, 4, 6, 8, 10} A = {2, 6} B = {0, 2, 6, 10} U B A The universal set contains all elements of interest.
  6. 6. Using Set Notation <ul><li>Example : </li></ul><ul><li>Let A = {2, 4, 6, 8, 10}, B = {2, 4, 6} and C = {2}. Write True or False to each statement. </li></ul>(a) B  A (b) B = C (c)  = C (d) A  B True. All elements in B are also in A . False. B and C do not have the same elements. True. The empty set is a subset of every set. False. There are elements in A that are not in B.
  7. 7. Classification of Numbers <ul><li>The natural numbers , or counting numbers , are the numbers in the set </li></ul>The whole numbers are the numbers in the set W = {0, 1, 2, 3, …}. Ellipsis indicate that the pattern continues indefinitely.
  8. 8. Classification of Numbers <ul><li>The integers are the numbers in the set </li></ul>The set of rational integers are the numbers in the set A rational number is a number that can be expressed as a quotient of two integers. The integer p is called the numerator , and the integer q is called the denominator .
  9. 9. Rational Numbers <ul><li>Examples of rational numbers are </li></ul>Rational numbers can also be represented as decimals. The decimal representation of a number is found by carrying out the division indicated. The repeat bar shows that the pattern continues. These decimals terminate .
  10. 10. Irrational Numbers <ul><li>An irrational number is a decimal that neither terminates or repeats. </li></ul>These decimals continue. Notice that if a number is rational, it cannot be irrational and vice-versa.
  11. 11. The Real Numbers Irrational numbers Rational numbers Integers Whole Numbers Natural Numbers
  12. 12. Classifying Numbers <ul><li>Example : </li></ul><ul><li>List the numbers in the set </li></ul><ul><li>that are </li></ul>(a) Counting numbers (b) Whole numbers (c) Integers (d) Rational numbers (e) Irrational numbers (f) Real numbers
  13. 13. Truncating Decimals <ul><li>To truncate a decimal, drop all the digits that follow the specified final digit in the decimal. </li></ul>Example : Approximate 45.28502 to two decimal places by truncating. 45.28502 45.28
  14. 14. Rounding Decimals <ul><li>To round a decimal, identify the final digit in the decimal. If the next digit is 5 or more, add 1 to the final digit; if the next digit is 4 or less, leave the final digit as it is, then truncate all digits to the right of the final digit. </li></ul>Example : Approximate 45.28502 to two decimal places by rounding. 45.28502 45.29 The 5 tells us to add 1 to the final digit.
  15. 15. The Real Number Line <ul><li>The real numbers can be represented by points on a line called the real number line . </li></ul>The distance between 0 and 1 determines the scale of the number line. The number associated with a point is called the coordinate . 2 – 2 0 1 3 4 5 – 1 – 3 – 4 – 5 Negative numbers Positive numbers Origin
  16. 16. Plotting Points Example : On the real number line, label the points with the coordinates 0, 1.75, – 2.5, 4. Real numbers are named in order on the number line. – 2.5 < 0 0 > 4 2 – 2 0 1 3 4 5 – 1 – 3 – 4 – 5 “ less than” “ greater than”
  17. 17. Ordering Numbers Example : Replace the ? With < , > or =. (a) 3 ? –2 Since 3 lies to the right of –2, we know that 3 > –2. (b) 3 ? 5.6 Since 3 lies to the left of 5.6, we know that 3 < 5.6. > < (c) – 2 ? Since – 2 is another way of writing we know that =
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