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2.3 Deductive Reasoning Deductive means “a systematic method of deriving conclusions” 2-3 Deductive Reasoning You will use symbolic notation to represent logical statements. You will learn to form conclusions by applying the laws of logic to the statements Symbolic Notation p represents the hypothesis q represents the conclusion → is read as “implies” ~ represents negation If p then q can be written as p →q Inverse statement ~p → ~q Bi-conditional statement p ↔ q How would you represent each of the following “symbolically?” Conditional pq Converse q p Inverse Contrapositive p q q p Deductive Reasoning Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument. Inductive reasoning uses previous examples and patterns to make a conjecture. Laws of Deductive Reasoning Law of Detachment: If p →q is a true conditional statement and p is true, that is, the situation described in the hypothesis occurs, then q is true, that is, the conditional situation also occurs. If it snows 10 feet, then there is no school. It just snowed 10 feet, so I can conclude there is no school. Laws of Deductive Reasoning Law of Syllogism: If p →q and q →r are true conditional statement, then p →r is true p: John gets a C q: John passes the test r: John plays football If John gets a C, then john plays football. Write the Converse, Inverse, and Contrapositive in symbolic notation. C. Statement: If it is Wednesday, then I am not home. pq Converse: If I am not at home, then it is Wednesday. q p Inverse: If it is not Wednesday, then I am home. p q Contrapositive: If I am home, then it is not Wednesday. q p Write the Converse, Inverse, and Contrapositive (include symbolic notation). If 3 measures 90, then 3 is not acute. pq If 3 is not acute, then 3 measures 90. q p If 3 does not measure 90, then 3 is acute. p q If 3 is acute, then 3 does not measure 90. q p