SPCR.1a – Lesson A Levels 1 – 3 Describe events as subsets of a sample space and use Venn diagrams to represent intersections, unions, and complements. Introduction to Venn Diagrams D. Otap – Fall 2016

Sets Definitions Set – a collection of items (elements) that have something in common or follow a rule. Element – individual items in a set Notation Set - Italicized Capital letter Element Italicized Lowercase letters Elements of a set are listed inside Curly Brackets and separated by commas

Each element of a set is unique should be listed more than once Sets   Each element of a set is unique So, no element should be listed more than once within the set

Remember - Each element of a set is unique Sets Examples Possible outcomes for flipping a coin: {heads, tails} Possible outcomes for rolling a standard 6-sided die {1, 2, 3, 4, 5, 6} Grade levels at Wando {Freshman, Sophomore, Junior, Senior} Remember - Each element of a set is unique

Venn Diagrams Definition Venn Diagram– a diagram which uses circles (or other closed shapes) to represent sets. The universal set is represented by a rectangle. Individual sets within the universal set are represented by circles (or other closed shapes) within the rectangle.  A U B A

Using Sets and Venn Diagrams Example Problem 1: We want to look at how many students in a class have at least 1 sister and/or have at least 1 brother.

Let’s Create a Venn Diagram Step 1. Draw the Universal Set

What is the Universal Set for this example? U = {all of the students in the class}

Step 2. Draw a circle for each possible outcome – they should overlap (We have 2 possible outcomes.)

Notice the overlap? This is the called the INTERSECTION. In this problem, the intersection represents students who have at least 1 brother and at least 1 sister.

Question 1 - What does section A represent Question 1 - What does section A represent? (Click on the correct answer.) Students with No Siblings Students with Sisters Only (No Brothers) Students with Brothers Only (No Sisters) Students with at least 1 Sister and at least 1 Brother

Not Quite While any student with a Sister goes into the yellow circle, only those with a sister, but no brother go into the section labeled A. If a student with a sister, also has a brother, the student would go into the section labeled B.

That’s right! Section A represents students with at least 1 sister and no brothers.

Students with at least 1 Sister and at least 1 Brother Question 2 - What does section B represent? (Click on the correct answer.) Students with No Siblings Students with Sisters Only (No Brothers) Students with Brothers Only (No Sisters) Students with at least 1 Sister and at least 1 Brother

Not Quite While any student with a Sister goes into the yellow circle and any student with a brother goes into the red circle, students with both sister(s) and brother(s) go into the intersection of the circles, section B.

That’s right! Section B represents students with at least 1 sister and at least 1 brother.

Question 3 - What does section C represent Question 3 - What does section C represent? (Click on the correct answer.) Students with No Siblings Students with Sisters Only (No Brothers) Students with Brothers Only (No Sisters) Students with at least 1 Sister and at least 1 Brother

Not Quite While any student with a Brother goes into the red circle, only those with a brother, but no sister go into the section labeled C. If a student with a sister, also has a brother, the student would go into the section labeled B.

That’s right! Section C represents students with at least 1 brother and no sisters.

Question 4 - What does the blue section (Section D) represent Question 4 - What does the blue section (Section D) represent? (Click on the correct answer.) Students with No Siblings Students with Sisters Only (No Brothers) Students with Brothers Only (No Sisters) Students with at least 1 Sister and at least 1 Brother

Not Quite Any students with a sister go in the yellow circle and any students with a brother go into the red circle. Students with no siblings go in the outer blue part, section D.

That’s right! The blue section (D) represents the students with no siblings.

Notation