1. 8C Truth Tables, 8D, 8E Implications 8F Valid Arguments
Unit 7: Logic and Sets
8C, 8D, 8E
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2. Truth Tables
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used to organize all cases for truth values number of rows: where n is the number of propositions tautology a statement that only has “True” truth values contradiction a statement that only has “False” truth values logically equivalent when two implications or compound propositions have identical truth values
8C, 8D, 8E
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3. Implications
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conditional statements “If …, then … ” p: I confiscate your cell phone. antecedent q: You ask for your phone at the end of the day. consequent : If I confiscate your cell phone, then you ask for it at the end of the day. p q T F p q T F
8C, 8D, 8E
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4. Related Implications
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given implication: converse: inverse: contrapositive: p: a figure is a rectangle q: a figure is a quadrilateral is true is false what does a truth table containing a statement and all its related implications indicate?
logically equivalent
logically equivalent
8C, 8D, 8E
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5. Implication and its Converse
r: The triangle is equilateral. s: The triangle is equiangular. is true “If the triangle is equilateral, then it is equiangular.” “If the triangle is equiangular, then it is equilateral.” equivalence (biconditional) “… if and only if …” “The triangle is equilateral if and only if it is equiangular.” When is an equivalence true?
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8C, 8D, 8E
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6. Implication and its Converse
p: The figure is a square. q: The figure is a triangle. is false “The figure is a square if and only if it is a triangle.” An equivalence is not the same as saying that two statements are logically equivalent to one another.
8C, 8D, 8E
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7. Formula Booklet
contains a reference for when negations, compound statements, implications, and equivalences are true or false
8C, 8D, 8E
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8. Validity vs. Truth
“If the sky is blue, then it is sunny.” “If it is sunny, then I am sad.” “If the sky is blue, then I am sad.” individual statements may be true or false, but conclusions reached using logical reasoning are either valid or invalid
8C, 8D, 8E
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9. Argument
made up by a set of propositions (premises) that leads to a conclusion All men are mortal. Socrates is a man. Hence, Socrates is mortal. Fifty percent of the students in this class are Republicans. Therefore, 50 percent of all students at this college are Republicans. inductive argument: intends to provide probable support for their conclusions cogent: inductive argument that provides good reasons for accepting the conclusion deductive argument: intends to provide conclusive support for their conclusions valid: refers to a deductive argument’s logical structure; argument is valid if “the conclusion must follow the premises”; a valid argument with true premises guarantees a true conclusion
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10. Form: Disjunctive Syllogism
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Either Jill faked the UFO landing or Jack did. Jill did not fake the UFO landing. Therefore, Jack faked the UFO landing. p: Jill faked the UFO landing. q: Jack faked the UFO landing. argument:
Valid! If an argument is a tautology, then it is valid. p q T F p q T F p q T F p q T F p q T F 8F
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11. Form: Denying the Antecedent
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If Joan is a widow, then Joan is female. Joan is not a widow. Therefore, Joan is not female. p: Joan is a widow. q: Joan is female. argument:
Invalid!
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12. Form: Affirming the Antecedent
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If Joe is a bachelor, then Joe is male. Joe is a bachelor. Therefore, Joe is male. p: Joe is a bachelor. q: Joe is male. argument:
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Valid! 8F
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13. Form: Affirming the Antecedent (Second Example)
If Joe is a bachelor, then = 5. Joe is a bachelor. Therefore, = 5. p: Joe is a bachelor. q: = 5. argument:
It is not a “sound” argument if the premises are not both true.
Valid! 8F
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14. Form: Hypothetical Syllogism
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If Joe is a bachelor, then Joe is male. If Joe is male, then Joe has a Y chromosome. Therefore, if Joe is a bachelor, then Joe has a Y chromosome. p: Joe is a bachelor. q: Joe is male. r: Joe has a Y chromosome. argument:
Valid!
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15. Guided Practice
p. 242: 2abd,4d,6 p. 244: 1a, 2c p. 246: 1c, 2a, 3ab, 4, 5cd, 6a p. 249: 1e, 2b, 6 p. 251: 1, 2a, 6abe p. 253: 1ab, 3, 4 Read and follow all instructions. List the page and problem numbers alongside your work and answers in your notes. Use the back of the book to check your answers.
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8C, 8D, 8E
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