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Lesson 2.4 Logical Sequencing & Conditional Statements Objective: Using logical sequencing and conditional statements, converse, inverse, and contrapositive.

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Logical sequencing is useful to know because it helps us to think logically through a problem, and put a solution in a form that everyone can understand and follow. (Think about your teacher explaining how to do a math problem to you…out loud…in words!) In order to do this we use logical sequences, and we must begin with conditional statements!!! Why are we doing this?

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Conditional Statements A conditional statement is written in the form if p, then q. If a given condition is met (if p), then another condition is true or an event will happen (then q). The if-clause is the hypothesis; the then- clause is the conclusion.

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Conditional Statements Ex 1.) If you don’t do your homework, then you will get a zero. Ex 2) Rewrite the statement in if-then form: 2.) Every multiple of 4 is also a multiple of 2. If a number is a multiple of 4, then it is a multiple of 2. True or False? True p - Hypothesis q - Conclusion

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Logical Order When given several related conditional statements, you must put them in logical order. The conclusion of one statement will flow into the hypothesis of the next. This is what we do in a paragraph proof!!!

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Logical Order – Put the following if-then statements in order A. If Cameron graduates with a degree, then he will make a lot of money. B. If Cameron studies hard, then his grades will be good. C. If Cameron makes a lot of money, then he will be able to buy a new car. D. If Cameron attends college, then he will graduate with a degree. E. If Cameron has good grades, then he will be able to attend college. Logical Order: B. If Cameron studies hard, then his grades will be good. E. If Cameron has good grades, then he will be able to attend college. D. If Cameron attends college, then he will graduate with a degree. A. If Cameron graduates with a degree, then he will make a lot of money. C. If Cameron makes a lot of money, then he will be able to buy a new car. Conclusion: If Cameron studies hard, then he will be able to buy a new car. This is an example of a SYLLOGISM – it is just a logical progression, you will do one on your homework

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Logical Order Ex. 2 – Put the statements in logical order. A. If a shape is a square, then it is a rhombus. B. If a shape is a parallelogram, then it is a quadrilateral. C. If a shape is a quadrilateral, then it is a polygon. D. If a shape is a rhombus, then it is a parallelogram. A. If a shape is a square, then it is a rhombus. D. If a shape is a rhombus, then it is a parallelogram. B. If a shape is a parallelogram, then it is a quadrilateral. C. If a shape is a quadrilateral, then it is a polygon. Conclusion: If a shape is a square, then it is a polygon.

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Converse The converse of a conditional statement is formed by switching the places of the hypothesis and conclusion. The sentence if p, then q becomes if q, then p. What is the converse? Is the given statement true or false? Write the converse statement for each conditional statement. Is the converse true or false? If the converse is false, come up with a counter example.

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Examples: 1. If a quadrilateral is a rectangle, then it is a parallelogram. TRUE If a quadrilateral is a parallelogram, then it is a rectangle. False It could be a rhombus

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Inverse The inverse of a conditional statement is formed by negating the hypothesis and the conclusion. The sentence if p, then q becomes if not p, then not q. What is the Inverse? Ex: If it is sunny outside, then I will go running. Becomes… If it is not sunny outside, then I will not go running.

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Is the given statement true or false? Write the inverse statement for each conditional statement. Is the inverse true or false? If the inverse is false, come up with a counter example. 4. Example: If two lines are perpendicular, then they intersect. TRUE False If two lines are not perpendicular, then they don’t intersect.

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Contrapositive The contrapositive of a conditional statement is formed in two steps. 1. Form the converse of the statement 2. Form the inverse of the converse. In other words, the sentence if p, then q becomes if not q, then not p. What is a contrapositive Statement Is the given statement true or false? Write the contrapositive statement for each conditional statement. Is the contrapositive true or false? If the contrapositive is false, come up with a counter example.

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Examples 7. If a polygon has just four sides, then it is a quadrilateral. If it is not a quadrilateral, then it is not a polygon with four sides. TRUE 8. If 2 angles form a straight line, then their sum is. TRUE If the sum is not, then the 2 angles don’t form a straight line. TRUE

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When will they be true: If your original is true then your _____________is usually true and usually your ____________and ___________ will be __________ contrapositive FALSE inverse converse

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When will they be true: If your original is false then your _____________is false and usually your ____________and ___________ will be __________ contrapositive TRUE inverse converse

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Homework Worksheet 2.4 and Syllogism Poster Your syllogism poster is easier than you think!!....

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Chapter 2: Reasoning and Proof Section 2.1 - Conditional Statements.

Chapter 2: Reasoning and Proof Section 2.1 - Conditional Statements.

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