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# Grade 9 - 1. Sets and Venn Diagrams

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by

Tweet## Steve Myers

on 11 September 2012#### Transcript of Grade 9 - 1. Sets and Venn Diagrams

Grade 9 IGCSE Math Extended Sets and Venn Diagrams Sets and Venn Diagrams Opening Problem: A baseball team of 21 players has 15 players that bat right handed and 6 that are ambidextrous. There are 3 pitchers who do not bat. How many players bat only left handed?

How many players bat only right handed? Word of the Day Set Notation Set - a collection of objects or numbers We represent a set with a capital letter OR with curly brackets {} For example: A = the set of all students in grade 9 at Riverside E = {0,2,4,6,8} R = {red, orange, yellow, green, blue, indigo, violet} Elements Elements are the objects or things that make up a set. - "is an element of" - "is NOT an element of" So, if , then and . Subsets Quick Check: P is the set of all odd numbers less than or equal to 13. List the elements of P.

True or false: A is a subset of B if all elements in A are also in B.

A is a proper subset of B if it is a subset of B, but NOT EQUAL to B. - "A is a subset of B."

- "A is a proper subset of B." What's the difference between subset and proper subset? The definition of subset says "If A is a subset of B, then all elements of A are also in B." By this definition, could A be exactly the same set as B? The definition of PROPER subset excludes the possibility of the two sets being equal. In other words, if . then If A = {1,2,3}, list all possible subsets of A. Which are proper subsets? For example:

If B = {-2,-1,0,1,2}, and A = {-2,0,2},

then . ** An element can only appear in a set ONCE. Don't write the same number or object twice. Ex: Don't write {2,0,1,2,2}. Simply write {0,1,2}. TWO IMPORTANT SETS: The universal set - the set of all elements under consideration.

The empty set - the set with no elements. - the universal set

- the empty set The Complement of a Set The complement of a set A is the set of all the elements in the universal set that are NOT in A. - the complement of A. No element in A can be in and no element in can be in A. So if B is the set of all boys in this classroom, then what would be? Take a few minutes to practice! Work on Exercise 2A on p. 59. PRACTICE Special Number Sets Natural numbers -

Integers -

Positive Integers -

Rational Numbers -

Real Numbers - the set of all

numbers on the number line (cc) image by nuonsolarteam on Flickr Set Builder Notation "The set of all m over n such that m and n belong to the set of integers." Example 1 Example 2 Take a few minutes to practice! Work on Exercise 2B on p. 61. PRACTICE Interval Notation We can easily describe sets of numbers using only symbols using interval notation: "The set of all real numbers x such that x lies between negative three and two, including two." In words: On the number line: If we want to talk about a different set of numbers, we can: Interval Notation (cont'd) In words: On the number line: "The set of all integers x such that x lies between negative five and five." Example 3 Example 4 PRACTICE Take a few minutes to practice! Work on Exercise 2C on pg. 62 in your textbook. Venn Diagrams A Venn diagram is a visual representation of sets. We use a rectangle for the universal set and circles for any other set. Example 5 Subsets: Circles Within Circles If A is a subset of B, we make a circle for A completely inside B: Intersecting Sets: Overlapping Circles If two sets have elements in common, but they are not subsets of each other, then we use overlapping circles: Example 6 Example 7 PRACTICE Take a few minutes to practice! Work on Exercise 2D on pg. 64 in your textbook. Union and Intersection The union of A and B is the set of all elements in A OR B OR both.

The intersection of A and B is the set of all elements in both A AND B. - the union of A and B. - the intersection of A and B. Union and Intersection Quick Check: List the elements of : a) b) Disjoint Sets Two sets are disjoint if they have no elements in common. In other words, if A and B are disjoint, then . (If they have elements in common, they are called non-disjoint.) What would two disjoint sets represented in a Venn Diagram look like? Example 8 Set Identities Big Idea: These identities are always true: We can prove these using Venn diagrams! Example 9 PRACTICE Take a few minutes to practice. Work on Exercises 2E.1 and 2E.2 starting on pg. 66 in your textbook. Problem Solving

With Venn Diagrams When problem solving, we use a number in brackets to represent the number of elements in a region. Example 10 Example 11 Example 12 PRACTICE Work on Exercise 2F on pg. 71 in your textbook. Last exercise for this topic! Woohoo! - the number of elements in the set A. and Interval Notation: Big Idea: Subsets and Proper Subsets

Full transcriptHow many players bat only right handed? Word of the Day Set Notation Set - a collection of objects or numbers We represent a set with a capital letter OR with curly brackets {} For example: A = the set of all students in grade 9 at Riverside E = {0,2,4,6,8} R = {red, orange, yellow, green, blue, indigo, violet} Elements Elements are the objects or things that make up a set. - "is an element of" - "is NOT an element of" So, if , then and . Subsets Quick Check: P is the set of all odd numbers less than or equal to 13. List the elements of P.

True or false: A is a subset of B if all elements in A are also in B.

A is a proper subset of B if it is a subset of B, but NOT EQUAL to B. - "A is a subset of B."

- "A is a proper subset of B." What's the difference between subset and proper subset? The definition of subset says "If A is a subset of B, then all elements of A are also in B." By this definition, could A be exactly the same set as B? The definition of PROPER subset excludes the possibility of the two sets being equal. In other words, if . then If A = {1,2,3}, list all possible subsets of A. Which are proper subsets? For example:

If B = {-2,-1,0,1,2}, and A = {-2,0,2},

then . ** An element can only appear in a set ONCE. Don't write the same number or object twice. Ex: Don't write {2,0,1,2,2}. Simply write {0,1,2}. TWO IMPORTANT SETS: The universal set - the set of all elements under consideration.

The empty set - the set with no elements. - the universal set

- the empty set The Complement of a Set The complement of a set A is the set of all the elements in the universal set that are NOT in A. - the complement of A. No element in A can be in and no element in can be in A. So if B is the set of all boys in this classroom, then what would be? Take a few minutes to practice! Work on Exercise 2A on p. 59. PRACTICE Special Number Sets Natural numbers -

Integers -

Positive Integers -

Rational Numbers -

Real Numbers - the set of all

numbers on the number line (cc) image by nuonsolarteam on Flickr Set Builder Notation "The set of all m over n such that m and n belong to the set of integers." Example 1 Example 2 Take a few minutes to practice! Work on Exercise 2B on p. 61. PRACTICE Interval Notation We can easily describe sets of numbers using only symbols using interval notation: "The set of all real numbers x such that x lies between negative three and two, including two." In words: On the number line: If we want to talk about a different set of numbers, we can: Interval Notation (cont'd) In words: On the number line: "The set of all integers x such that x lies between negative five and five." Example 3 Example 4 PRACTICE Take a few minutes to practice! Work on Exercise 2C on pg. 62 in your textbook. Venn Diagrams A Venn diagram is a visual representation of sets. We use a rectangle for the universal set and circles for any other set. Example 5 Subsets: Circles Within Circles If A is a subset of B, we make a circle for A completely inside B: Intersecting Sets: Overlapping Circles If two sets have elements in common, but they are not subsets of each other, then we use overlapping circles: Example 6 Example 7 PRACTICE Take a few minutes to practice! Work on Exercise 2D on pg. 64 in your textbook. Union and Intersection The union of A and B is the set of all elements in A OR B OR both.

The intersection of A and B is the set of all elements in both A AND B. - the union of A and B. - the intersection of A and B. Union and Intersection Quick Check: List the elements of : a) b) Disjoint Sets Two sets are disjoint if they have no elements in common. In other words, if A and B are disjoint, then . (If they have elements in common, they are called non-disjoint.) What would two disjoint sets represented in a Venn Diagram look like? Example 8 Set Identities Big Idea: These identities are always true: We can prove these using Venn diagrams! Example 9 PRACTICE Take a few minutes to practice. Work on Exercises 2E.1 and 2E.2 starting on pg. 66 in your textbook. Problem Solving

With Venn Diagrams When problem solving, we use a number in brackets to represent the number of elements in a region. Example 10 Example 11 Example 12 PRACTICE Work on Exercise 2F on pg. 71 in your textbook. Last exercise for this topic! Woohoo! - the number of elements in the set A. and Interval Notation: Big Idea: Subsets and Proper Subsets