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1/15/201412.1: Truth and Validity in Logical Arguments Expectations: L3.2.1: Know and use the terms of basic logic L3.3.3: Explain the difference between a necessary and a sufficient condition within the statement of a theorem.

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1/15/201412.1: Truth and Validity in Logical Arguments Logical Argument Statements Conclusion – final statement Premises – all statements preceding the conclusion

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1/15/201412.1: Truth and Validity in Logical Arguments Valid Argument An argument is considered valid if the conclusion follows logically from the premises.

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1/15/201412.1: Truth and Validity in Logical Arguments Valid Reasoning If the premises are all true, then the conclusion will be ______.

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1/15/201412.1: Truth and Validity in Logical Arguments What conclusion follows from the premises: If a polygon is a square, then it is a rectangle. If a polygon is a rectangle, then it is a parallelogram. ABCD is a square.

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1/15/201412.1: Truth and Validity in Logical Arguments Is this a valid conclusion? Some triangles are isosceles. ABC is a triangle. Conclude: ABC is an isosceles triangle.

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1/15/201412.1: Truth and Validity in Logical Arguments Types of Arguments Modus Ponens: The Law of Detachment If (p => q) is a true conditional statement and p is a true statement, then ___________________________. This is a valid form of reasoning.

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1/15/201412.1: Truth and Validity in Logical Arguments Affirming the Consequent If (p => q) is a true conditional statement and q is a true statement, then p must be true. This is _____ a valid form of reasoning.

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1/15/201412.1: Truth and Validity in Logical Arguments Determine if the following conclusion is valid or invalid. If 2 lines are parallel, then they do not intersect. l does not intersect m. Conclude: l is parallel to m

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1/15/201412.1: Truth and Validity in Logical Arguments Determine if the following conclusion is valid or invalid. If a triangle is a right triangle, then it has a right angle. ΔABC is a right triangle. Conclude: ΔABC has a right angle.

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1/15/201412.1: Truth and Validity in Logical Arguments More Types of Arguments Modus Tollens: Law of the Contrapositive If p => q and ~q are true, then __________________ This is a ________ form of reasoning.

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1/15/201412.1: Truth and Validity in Logical Arguments Denying the Antecedent If p => q and ~p, then _____________. This is a ___________ valid form of reasoning.

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1/15/201412.1: Truth and Validity in Logical Arguments Determine if the following conclusion is valid or invalid. If x = 4, then x 2 = 16. x 2 16 Conclude: x 4

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1/15/201412.1: Truth and Validity in Logical Arguments Determine if the following conclusion is valid or invalid. If x = 3, then x 2 = 9. x 3 Conclude: x 2 9

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1/15/201412.1: Truth and Validity in Logical Arguments Necessary and Sufficient Conditions In the statement of a theorem in if- then form, we can talk about sufficient conditions for the truth of the statement and necessary conditions of the truth of the statement. This is really just another way of looking at the Law of Detachment and Affirming the Consequent.

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1/15/201412.1: Truth and Validity in Logical Arguments The ___________ is a sufficient condition for the conclusion and the ___________ is a necessary condition of the hypothesis.

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1/15/201412.1: Truth and Validity in Logical Arguments Necessary Consider the statement p => q. We say q is a necessary condition for (or of) p. Ex: If if is Sunday, then we do not have school. A necessary condition of it being Sunday is that we do not have school, but it is not sufficient to say it must be Sunday if we do not have school.

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1/15/201412.1: Truth and Validity in Logical Arguments Sufficient Condition A sufficient condition is a condition that all by itself guarantees another statement must be true. Ex: If you legally drive a car, then you are at least 15 years old. Driving legally guarantees that a person must be at least 15 years old.

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1/15/201412.1: Truth and Validity in Logical Arguments If M is the midpoint of segment AB, then AM MB. Given that M is the midpoint, it is necessary (true) that AM MB. This means that M being the midpoint is a ____________ condition for AM MB.

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1/15/201412.1: Truth and Validity in Logical Arguments Notice simply saying AM MB does not guarantee that M is the midpoint of AB, so it is not a sufficient condition.

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1/15/201412.1: Truth and Validity in Logical Arguments If a triangle is equilateral, then it is isosceles. A triangle having 3 congruent sides (equilateral) guarantees that at least 2 sides are congruent, so a triangle being equilateral is sufficient to say it is isosceles.

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1/15/201412.1: Truth and Validity in Logical Arguments If a person teaches mathematics, then they are good at algebra. Because Trevor is a math teacher, can we conclude he is good at algebra. Justify your answer.

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1/15/201412.1: Truth and Validity in Logical Arguments If a person teaches mathematics, then they are good at algebra. Betty is 32 and is very good at algebra. Can we correctly conclude that she is a math teacher? Justify.

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Which is not a sufficient condition for 2 lines being coplanar? A.they are parallel B.they are perpendicular C.they intersect D.they have no common points E.they have 2 common points 1/15/201412.1: Truth and Validity in Logical Arguments

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Which of the following is a necessary but not sufficient condition for angles to be supplementary? A.they form a linear pair. B.their angle measures add to 180. C.they are both right angles. D.their angle measures are 135 and 45. E.none of the above. 1/15/201412.1: Truth and Validity in Logical Arguments

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1/15/201412.1: Truth and Validity in Logical Arguments Bi-Conditional Statements If a statement and its converse are both true it is called a bi-conditional statement and can be written in ________________ form.

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1/15/201412.1: Truth and Validity in Logical Arguments Ex: If an angle is a right angle, then its measure is exactly 90° and If the measure of an angle is exactly 90°, then it is a right angle are true converses of each other so they can be combined into a single statement.

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1/15/201412.1: Truth and Validity in Logical Arguments Necessary and Sufficient If a statement is a bi-conditional statement then either part is a necessary and sufficient condition for the entire statement. Remember all definitions are bi-conditional statements.

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1/15/201412.1: Truth and Validity in Logical Arguments A triangle is a right triangle iff it has a right angle. Being a right triangle is necessary and sufficient for a triangle to have a right angle and possessing a right angle is necessary and sufficient for a triangle to be a right triangle.

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1/15/201412.1: Truth and Validity in Logical Arguments Necessary, Sufficient, Both or Neither Given the true statement: If a quadrilateral is a rhombus (4 congruent sides), then its diagonals are perpendicular. Is the following statement necessary, sufficient, both or neither? The diagonals of ABCD are perpendicular.

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1/15/201412.1: Truth and Validity in Logical Arguments Necessary, Sufficient, Both or Neither Given the true statement: A quadrilateral is a rhombus if and only if its 4 sides are congruent. Is the following statement necessary, sufficient, both or neither? The sides of ABCD are all congruent.

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1/15/201412.1: Truth and Validity in Logical Arguments Assignment pages 772 – 774, # 7-15 (odds), 21-34 (all)

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