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**2.3**DEDUCTIVE REASONING 1 2 GOAL GOAL Use symbolic notation to represent logical statements Form conclusions by applying the laws of logic to true statements. The laws of logic help you with classification, such as determining true statements about different birds. Whatyou should learn Why you should learn it**2.3**DEDUCTIVE REASONING USING SYMBOLIC NOTATION 1 GOAL You have already learned about conditional statements in if-then form: If HYPOTHESIS, then CONCLUSION. Now we can use symbolic notation to show and work with the same ideas.**Conditional statement: If p, then q or p**q. If q, then p or q p. p if and only if q or p q. EXAMPLE 1 If HYPOTHESIS, then CONCLUSION. Letting p represent the hypothesis and q represent the conclusion, we can write “If p, then q,” or “p q.” Using this notation we can also write the converse and a biconditional statement in symbolic notation. Converse: Biconditional: The biconditional statement may also be written as If p, then q and if q, then p.**a. Write in words.**b. Write in words. Since Since means “If q, then p,” we write means “If p, then q,” we write c. Decide whether the biconditional statement is true. q p q p q q p q p p p q Although p is true, q is false (x could be 4), so the biconditional statement is false. Extra Example 1 Let p be “the value of x is –4” and q be “the square of x is 16.” Solution: If the value of x is –4, then the square of x is 16. Solution: If the square of x is 16, then the value of x is –4. Solution:**If not p, then not q or ~p ~q.**If not q, then not p or ~q ~p. EXAMPLE 2 Negation Since writing the inverse and contrapositive require negation, we need a symbol for negation: That symbol is “~”. Be certain you review the examples at the top of page 88 for more explanation! Inverse: Contrapositive:**~q ~p**~p ~q q q p p Extra Example 2 • Let p be “today is Monday” and q be “there is school.” • Write the contrapositive of in symbols and words. • b. Write the inverse of in symbols and words. If there is no school, then today is not Monday. If today is not Monday, then there is no school.**No; is not true.**q q p q q q p p p q p p Checkpoint • Let p be “a number is divisible by 3” and q be “a number is divisible by 6.” • Write in words. • Write in words. • Decide whether the biconditional statement is true. • Write the contrapositive of . • Write the inverse of . If a number is divisible by 3, then it is divisible by 6. If a number is divisible by 6, then it is divisible by 3. If a number is not divisible by 6, then it is not divisible by 3. If a number is not divisible by 3, then it is not divisible by 6. Do you remember which are equivalent statements? Review the chart on the bottom of page 88.**2.3**DEDUCTIVE REASONING 2 GOAL USING THE LAWS OF LOGIC EXAMPLE 3 • DEDUCTIVE REASONING • Uses facts definitions and accepted properties in a logical order to write a logical argument. • This is NOT THE SAME as inductive reasoning, which uses previous examples and patterns to form a conjecture. REMEMBER THE DIFFERENCE!**Extra Example 3**Is the reasoning inductive or deductive? a. Josh knows that Brand X computers cost less than Brand Y computers. All other brands that Josh knows of cost less than Brand X. Josh reasons that Brand Y costs more than all other brands. Inductive (it’s based on past observations). b. Josh knows that Brand X computers cost less than Brand Y computers. He also knows that Brand Y computers cost less than Brand Z. Josh reasons that Brand X costs less than Brand Z. Deductive (it’s based on facts).**LAW OF DETACHMENT**If is a true conditional statement and p is true, then q is true. EXAMPLE 4 q p Laws of Deductive Reasoning**If two angles are vertical, then they are congruent.**and are vertical. So and are congruent. Extra Example 4 State whether the argument is valid. Michael knows that if he does not do his chores in the morning, he will not be allowed to play video games later the same day. Michael does not play video games on Friday afternoon. So Michael did not do his chores on Friday morning. This is not a valid argument. (Michael could have done his chores and still chosen not play video games.) This is a valid argument; it follows the Law of Detachment.**If then is an acute**angle. So is an acute angle. Checkpoint State whether the argument is valid. • Sarah knows that all sophomores take driver education in her school. Hank takes driver education. So Hank is a sophomore. not valid valid**LAW OF SYLLOGISM**If and are true conditional statements, then is true. EXAMPLE 5 q r r p q p Laws of Deductive Reasoning**Extra Example 5**Write some conditional statements that can be made from the following true statements using the Law of Syllogism. • If a fish swims at 68 mi/h, then it swims at 110 km/h. • If a fish can swim at 110 km/h, then it is a sailfish. • If a fish is the largest species of fish , then it is a great white shark. • If a fish weighs over 2000 lb, then it is the largest species of fish. • If a fish is the fastest species of fish, then it can reach speeds of 68 mi/h.**EXAMPLE 6**Extra Example 5 (cont.) Sample answers. • If a fish swims at 68 mi/h, then it is a sailfish. (1 and 2) • If a fish is the fastest species of fish, then it is a sailfish. (1, 2, and 5) • If a fish weighs over 2000 lb, then it is a great white shark. (3 and 4) Others???**Let p be “Casey goes to a music store.” Let q be**“Casey shops for a CD.” Let r be “Casey will buy a CD.” Since and are both true, by the Law of Syllogism is true. Since p is true (Casey goes to a music store), by the Law of Detachment q (Casey will buy a CD) must also be true. r q r p p q Extra Example 6 Casey goes to a music store. Given the following true statements, can you conclude that Casey buys a CD? If Casey goes to a music store, she shops for a CD. If Casey shops for a CD, then Casey will buy a CD. Yes, Casey buys a CD. Explanation:**Checkpoint**Write a conditional statement that can be made from the following true statements using the Law of Syllogism. If a plant is the largest plant on Earth, then the plant is a Sierra Redwood tree. If a plant is a Sierra Redwood tree, then the plant can weigh 1,800,000 kg. If a plant is the largest plant on Earth, then the plant can weigh 1,800,000 kg.