Let s start by defining what it means for two statements to be logically equivalent.

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2 Boolean Algebra Boolean Algebra ( ) So far we have been presented with statements such as p, q, p, p q, p q, etc. It would be nice if we had a means of manipulating these statements in an algebraic manner. This would allow us to show the equality of two statements without having to build truth tables. Luckily someone solved the problem of creating a method of manipulation, and we just need to learn and understand it. Let s start by defining what it means for two statements to be logically equivalent. Definition of Logically Equivalent Two statements are logically equivalent,, or equal, =, iff they have the same logical content. Note: We typically use the symbol for equivalence. For the purpose of this lesson, we will not bother ourselves with the difference between = and and just use = since we should be familiar with =. Using a truth table, we can visually represent logical equivalence. Ex 1: Show that p q = p q. (Or we could say p q p q.) Solution: p q p p q p q T T F T T T F F F F F T T T T F F T T T Example 1 By observing the table, we see that Column p q is identical to Column p q. Hence p q is equal to, or logically equivalent to, p q. It s great that we can show logical equivalence with truth tables, but can we do it without? First we will need to learn a few properties of Boolean algebra. Property 0: Disjunction of Conjunctions Any compound statement of the form p q where represents a connective can be written as a disjunction of conjunctions. For example, p q is in that form.

3 Boolean Algebra Ex 10: State the properties between each step that prove p q = ( p q) ( q p). The proof is given below. p q = Disjunction of Conjunctions = ( p q) ( p q) Distributivity ( ) ( ) = p p q q p q Distributivity = ( p p) ( p q) ( q p) ( q q) A property of negation = TRUE ( p q) ( q p) TRUE Identity = ( p q) q p DeMorgan s Laws = ( p q) ( q p) ( ) Example 10 Ex 11: Prove that p q = p q without using a truth table. Solution: p q= We can use Property 0: Disjunction of Conjunctions. (Alternatively, we can use the work we showed in Example 2.) = ( p q) ( p q) p q ( ) Associativity will allow us to make the grouping shown in the next step. Example 11 = ( p q) ( p q) ( p q) Distributivity (in the reverse direction) allows us to factor out p from ( q q). = ( p q) p ( q q) We can replace q q property we learned with negations. [ ] = ( p q) p TRUE ( ) with TRUE because of a

4 Advanced Mathematics Radian Measure Degree Writing Assignment Degree Writing Assignment In order to fully understand radian measure, let s first look at degree measure and understand its origins. Create a fully developed response to the following two topics. 1.) Research theories about the origin of degrees, the use of splitting a circle into 360 degrees, and the motivation of ancient mathematicians to study angles. 2.) Discuss if dividing a circle into a number of pieces other than 360 would be more efficient. Why not use the number 720 or 600 (or any other number)? What about using a number other than 360 is different when it comes to measuring angles?

5 Advanced Mathematics for Matrix Capstone: Encryption BTWPSNRXG.IZCMBC!XZTRKUM _WI You may not have realized, but the title of this project is actually an encrypted message. Your assignment will be to encrypt and decrypt a message while working with a partner. Before beginning, you ll need to know the process used to encrypt the phrase. Here s how it works: Encryption 1.) Write down a message, and then split the message into sets of 3 by 1 matrices. SS EE EE SAVE ME! AA! VV MM 2.) Using the key, located below, now transform the message into numbers. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z (space).?! S E E A! V M ) Multiply each matrix by another matrix that has an inverse. Remember, to check if an inverse exists, simply find the determinant det = -1 The determinant is nonzero; therefore, an inverse exists = = =

6 Plane Analytic Geometry Plane Analytic Geometry Introduction Although it may seem like a simple concept, there is much to be said about lines and curves on the plane. This lesson begins by briefly discussing the different forms in which a line can appear then explores plane analytic geometry and families of lines. Line Forms Tip: Before trying to memorize the following forms in which a line can be written, just consider the title of the line form. For example, the slope intercept form should contain a representation of slope and an intercept. Point Slope Form Let ( x1, y1) be a point and m be the slope of a line. Then the point slope form of a line is ( y y ) = mx ( x ) 1 1 A limitation of point-slope form is that vertical lines cannot be represented using it since vertical lines have no slope. Two-Point Form Let ( x1, y1) and ( x2, y2) be the points. Then the two point form of a line is y y1 y2 y1 = x x x x A limitation of two point form is that vertical lines cannot be represented using it since that would result in a denominator of zero on the right side of the equation above. Slope Intercept Form Let (,) 0 b be a point and m be the slope of a line. Then the slope intercept form of a line is y = mx+ b A limitation of point-slope form is that vertical lines cannot be represented using it since vertical lines have no slope.

7 Plane Analytic Geometry Intercept Form Let ( a, 0) be the x intercept and (,) 0 b be the y intercept. Then the intercept form of a line is x y + =1 a b The limitations of intercept form are that neither a nor b can be zero. Hence, intercept form cannot represent lines through the origin. Also, since a and b exist, neither horizontal lines nor vertical lines can be represented. In other words, if we had a horizontal line, it would not cross the x-axis, therefore we would have no x intercept ( a, 0 ). General Form The general form of a line is Ax + By + C = 0 such that both A 0 and B 0 (simultaneously). The general form can be used to represent all lines. Exploring the Equations of the Line Using the chart on the following page, show the equivalences of the different forms of the equation of the lines and discuss difficulties of going from one form to another due to the limitations of each form. As an example, two of the sections have already been completed.

8 Plane Analytic Geometry Slope Intercept Form y = mx+ b Point Slope Form ( y y1) = mx ( x 1) Two Point Form y y1 y2 y1 = x x x x Intercept Form x y + =1 a b General Form Ax By C + + = 0 Slope Intercept Form Point Slope Form y = mx+ b ( y y1) = mx ( x 1) Solutions for this column are on page a. Solutions for this column are on page b. Two Point Form y y1 y2 y1 = x x x x Intercept Form x y + =1 a b Solutions for this column are on page c. Solutions for this column are on page d. General Form Ax By C + + = 0 Solutions for this column are on page e.

9 Hyperbolic Modeling The Engineering Design Process The Steps: 1. Identify the Problem -Students will learn how to interpret information and write a concise, but informative problem statement. 2. Research the Problem - Students will learn to interpret information from articles. Students will have to find applicable articles and extract pertinent information. Students will have open discussions and sharing opportunities which will drive the learning of the fundamental content. 3. Brainstorm Solutions -Students will learn to plan ahead. They will think of various solutions to the problem and organize their ideas accordingly. 4. Choose a Solution - Students will use their critical thinking skills and the information they learned in Step 2 to choose the best solution to the problem. 5. Create and Develop a Prototype - Students will begin to build the prototype of their design. This will put their construction skills to the test as well as help them realize how something they design on paper and in their minds can be built in the physical world. 6. Test and Evaluate the Prototype - Students will see if what they built performs in the manner they intended. 7. Improve and Redesign - Students will reflect on the design and determine ways they can improve upon it. In addition to the engineering design process steps, the students will learn the overarching themes of the process: iteration, communication, imagination, and creativity. Students will learn that design is never completed and the process is based on iteration so that improvements can be made. Students will be challenged to maintain communication with not only their teammate(s), but the instructor and other classmates as well. Emphasis will be placed on imagination and creativity such that students are able to express themselves through the projects. 2 For more information visit

10 Hyperbolic Modeling Step 3: Brainstorm Solutions Brainstorm 3 possible solutions. If needed refer back to your problem statement to recall exactly what you need to do. Keep in mind the materials you can use for constructing your prototype. You may need to revisit the Research the Problem section to research the materials that may be used and then return to this step. Use the following three pages to sketch your solutions and write a description about each one. In the description, include the materials you plan to use, how the parts are joined together, and advantages/disadvantages of each design. Criteria to Include in Each Solution A hyperbolic cross section of the hyperboloid structure (labeled with the equation of the hyperbola). A sketch of the structural components of the design. Label what each material used for the prototype is and what it would represent if an actual structure were built. A small description that includes the functions of that design. Remember: The engineering design process is iterative and you may need to return to previous steps before moving forward. Teacher Note: If this project is being constructed in class, the following could be available. Note that materials and tools could be added to this list or taken away to change the complexity of the project. Materials 1. Paper Mache (1:1 ratio of white glue to water; newspaper) 2. Dowel Rods 3. Something to form a sturdy base (play dough, modeling clay, etc.) 4. An adhesive to force the dowel rods to stay together (hot glue is recommended because of its fast drying properties) 5. Paint 6. Optional: Foam board for a surface to mount/place the final design. Teacher Note: There is a second list of materials located in the "Test and Evaluate the Prototype" section that will be needed to do the Water Temperature test. 6 For more information visit

11 Hyperbolic Modeling Teacher Note: Alternately, the students could create a technical report. Presentation You will now present on your design. Create a presentation that addresses each step in the engineering design process. Included below are tips for things you may want to include in the presentation. Title Slide - Give your project a name and include the names of the participants in your group. Other slides (you do not have to limit yourself to one slide per step or the questions below): Identify the Problem - This could be a slide that outlines the objective of your project. Research the Problem - What did you research and why did you choose to research those topics? What did you find interesting in the research? Brainstorm Solutions/Choose a Solution- Include sketches of your proposed solutions. Have a little bit of information about each one. What made each design unique? Why did you not choose certain designs? Why did you choose the design you chose? If you combined parts of several designs, which parts did you combine and why? Maybe you chose a completely different design, explain why. Create and Develop a Prototype - Show off pictures of your final prototype. Describe the different parts. Cost Analysis - Include information about the cost of the different aspects outlined in the "Cost Analysis" section. Test and Evaluate the Prototype - Discuss how the prototype performed on the test. Improve and Redesign - How can you improve the design? Include recommendations on how to improve the design and how you think these improvements will affect the performance. 19 For more information visit

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