Foundations 30 Mrs. Wirz

 Introduction – Venn Diagrams Students that play sports Students that have a part-time job All students in Mrs. Wirz’s class

 Set: A collection of distinguishable objects for example, the set of whole numbers is w={1,2,3,…}  Element: An object in a set, for example 3 is an element of D, the set of all digits.  Universal Set: The set of all elements being considered, for example the universal set of digits is D={1,2,3,4,5,6,7,8,9}  Subset: A set whose elements all belong to another set, for example, the set of odd digits, O={1,3,5,7,9} is a subset of D. In set notation this is written as O D.

 Complement: All the elements that belong to a universal set, but do not belong to a subset. For example, the complement of subset O is O’={2,4,6,8}, which happens to be all the even digits.  Empty set: A set with no elements. The set is denoted as { } or.  Disjoint: Two or more sets having no elements in common; for example, the set of even numbers and the set of odd numbers are disjoint.

A. List the elements of the universal set of students in Foundations 30.

F SW

 F is the universal set.  W and S are subsets of F. Write W in set notation.  W’ is the complement of W  Describe what W’ contains  Write W’ in set notation What would make an empty set?

Example 1: A. Using set notation, define the universal set, S, of all natural numbers 1 to 500. B. Using set notation define, the subsets, F and T, which indicate the multiples of 5 and 10, from 1 to 500. C. Represents the sets and subsets using a Venn diagram.

Complete example 2 and 3 on handout.

 Finite Set: A set with a countable number of elements; for example, the set of even numbers less than 10, E+{2,4,6,8} is finite.  Infinite Set: A set with infinite number of elements; for example the set of natural numbers, N={1, 2, 3,…}  Mutually Exclusive: Two or more events that cannot occur at the same time; for example, the Sun rising and the Sun setting are mutually exclusive events.

 Text: page Do # 2,5,8,11

 Sets that are not disjoint share common elements.  Each area of a Venn diagram represents something different.  When two non-disjoint sets are represented in a Venn diagram, you can count the elements in both sets by counting the elements in each region only once.

 Each element in a universal set appears only once in a Venn diagram.  If an element occurs in more than one set, it is placed in the area of the Venn diagram where the sets overlap.

 Page (on the back of this handout) # 1, 3

U A B

U AB

 A\B is read as “ A minus B.” It denotes the set of elements that are in set A but not in set B.  Do Example 1

 fMhK2g0 fMhK2g0  Page , #1, 6,8,11

 Computer Science Software Internet searches Networking Programing  Electrical Engineering  Geometry and Topology  Biology, Physics, and Chemistry

 Do examples “Investigate the Math,” example 1,2, and 3.  Do Questions #2,4,6, and 9

 Conditional Statement: An “if-then” statement. example: “If it is raining outside, then we practice indoors.” Hypothesis: “It is raining outside.” Conclusion: “We practice indoors.” The hypothesis is the statement that follows “if,” and the conclusion follows “then.”

 Case 1: It rains outside, and we practice indoors. When the hypothesis and conclusion are both true, the conditional statement is true.  Case 2: It does not rain outside, and we practice indoors. When the hypothesis and the conclusion are both false, then the conditional statement is true.  Case 3: It does not rain outside, and we practice outdoors. When the hypothesis is false and the conclusion is true, the conditional statement is true.  Case 4: It rains outdoors, and we practice outdoors. When the hypothesis is true and the conclusion is false, then the conditional statement is false.

Let p represent the hypothesis and let q represent the conclusion. pq True False True FalseTrue False

 Converse: A conditional statement in which the hypothesis and the conclusion are switched. example: If the conditional statement is: If it is raining outside, then we will practice indoors. The converse would be: If we practice indoors, then it is raining outside. A counter-example could disprove this statement. Such as, you could be practicing indoors because the field is being used or under repair.

 Read over the “in summary.”  Do questions 2, 3, 4, 5, 6, 10, and 13 from pages

 Inverse: A statement that is formed by negating both the hypothesis and the conclusion of a conditional statement;  for example, for the statement “If a number is even then it is divisible by 2,” the inverse is “If a number is not even, then it is not divisible by 2.”

 Contrapositive: A statement that is formed by negating both the hypothesis and the conclusion of the converse of a conditional statement;  For example, for the statement “If a number is even, then it is divisible by 2,” the contrapositive is “If a number is not divisible by 2, then it is not even.”

 Mrs. Wirz said, “If a polygon is a triangle then it has three sides.” A. Is this statement true? Explain. B. Write the converse of this statement. Is it true? Explain. C. Write the inverse of this statement. Is it true? Explain. D. Write the contrapositive of this statement. Is it true? Explain.

 Consider the statement, “If you live in Saskatoon, then you live in Saskatchewan.” A. Is this statement true? Explain. B. Write the converse of this statement. Is it true? Explain. C. Write the inverse of this statement. Is it true? Explain. D. Write the contrapositive of this statement. Is it true? Explain.

 If a conditional statement is true, then its contrapositive is true, and vice versa.  If the inverse of a conditional statement is true, then the converse of the statement is also true, and vice versa.

Examples – Read to understand examples 1 and 2. Pay special attention to the notation boxes. Complete the Your Turns for example 1 and 2. Assignment - #1-8, page

 Review Assignment page 220, #1,2,3,4,6,8. Due: Monday September 24  Test – Tuesday September 25