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Sec 2-3 Concept: Deductive Reasoning Objective: Given a statement, use the laws of logic to form conclusions and determine if the statement is true through completion of daily work

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**a. Diego bought a pretzel**

Example 1: Use the true statements to determine whether the conclusion is true or false. Explain your reasoning If Diego goes shopping, then he will buy a pretzel If the mall is open, then Angela and Diego will go shopping If Angela goes shopping, then she will buy a pizza The mall is open a. Diego bought a pretzel TRUE! Since the mall is open, Angela and Diego go shopping and therefore, Diego buys a pretzel

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**Since the mall is open, Angela and Diego went shopping**

Example 1 cont.: Use the true statements to determine whether the conclusion is true or false. Explain your reasoning If Diego goes shopping, then he will buy a pretzel If the mall is open, then Angela and Diego will go shopping If Angela goes shopping, then she will buy a pizza The mall is open b. Angela and Diego went shopping TRUE! Since the mall is open, Angela and Diego went shopping

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**c. Angela bought a pretzel**

Example 1 cont.: Use the true statements to determine whether the conclusion is true or false. Explain your reasoning If Diego goes shopping, then he will buy a pretzel If the mall is open, then Angela and Diego will go shopping If Angela goes shopping, then she will buy a pizza The mall is open c. Angela bought a pretzel FALSE! Since the mall is open, Angela and Diego went shopping, therefore, Angela bought a pizza

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**d. Diego had some of Angela’s Pizza**

Example 1: Use the true statements to determine whether the conclusion is true or false. Explain your reasoning If Diego goes shopping, then he will buy a pretzel If the mall is open, then Angela and Diego will go shopping If Angela goes shopping, then she will buy a pizza The mall is open d. Diego had some of Angela’s Pizza FALSE! Since the mall is open, Angela and Diego went shopping, therefore, Diego bought a pretzel

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**Example 2: Deductive Reasoning**

B.

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**Example 3: Write the symbolic statement in words.**

p: the sky is cloudy q: it is raining 1. ~p The sky is not cloudy 2. ~q It is not raining 3. p→q If the sky is cloudy, then it is raining 4. ~p→~q 5. q→p 6. ~q→~p If the sky is not cloudy, then it is not raining. If it is raining, then the sky is cloudy If it is not raining, then the sky is not cloudy 7. p↔q The sky is cloudy if and only if it is raining.

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**p q q r r p LAW OF SYLLOGISM**

Example 4: Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. p q If you are a teenager, then you are always right If you are always right, then people will listen to you If you are a teenager, then people will listen to you q r r p Law of Detachment: If p→q is a true conditional statement and p is true, then q is true Law of Syllogism: If p→q and q→r, then p→r LAW OF SYLLOGISM

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**p q true Law of Detachment p: true q: must be true**

Example 4 Continued: Determine if statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. p q true If an angle is acute, then it is not obtuse <ABC is actue <ABC is not obtuse p: true q: must be true Law of Detachment: If p→q is a true conditional statement and p is true, then q is true Law of Syllogism: If p→q and q→r, then q→r Law of Detachment

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