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Unit 2: Proofs. Conditional Statements. Conditional statement is the mathy term for “if-then ” statement. Example: If a car is a Wrangler, then it is a Jeep. In a conditional, the part following the word “if” is the hypothesis . Hypothesis: a car is a Wrangler
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Conditional Statements • Conditional statement is the mathy term for “if-then” statement. • Example: If a car is a Wrangler, then it is a Jeep. • In a conditional, the part following the word “if” is the hypothesis. • Hypothesis: a car is a Wrangler • ** Leave the “If” off ** • The part following the word “then” is the conclusion. • Conclusion: it is a Jeep • ** Leave the “then” off ** • In logical notation, conditionals are written as follows: If p, then q or p => q (which reads as “p implies q.”)
Euler (pronounced “Oiler”) Diagrams • Create a Euler diagram for the previous conditional: • “If a car is a Wrangler, then it is a Jeep.”
More Euler • Consider the following statement: Timmy’s car is a Wrangler. • Where should we place Timmy’s car in the diagram? • By placing Timmy’s car into the Wrangler category on the Euler diagram, you can see that it is also included in the Jeep category.
Now let’s write it using words… Logical Argument: • If a car is a Wrangler, then it is a Jeep. • Timmy’s car is a Wrangler. • Therefore, Timmy’s car is a Jeep. This process is known as deductive reasoning, or deduction.
Example • Draw a Euler diagram that conveys the following information: If a student has Mr. Brilliandt for English, then they are student at GMC. Susan has Mr. Brilliandt for English. • What conclusion can you draw about Susan?
Reversing Conditionals • When you interchange the hypothesis and the conclusion of a conditional, the new conditional is called the converse of the original. • Example: Conditional: If a car is a Civic, then it is a Honda. Converse: If a car is a Honda, then it is a Civic. • The original conditional statement is true! • Is the converse? • Is there an example of a Honda that is not a Civic? • If so, then the converse is false! • An example that proves a statement is false is called a counterexample. • What is a counterexample for the given converse?
Practice • Write a conditional with the hypothesis “the polygon has four congruent sides” and the conclusion “the polygon is a square.” • Write the converse. • Is the conditional true? If no, what is a counterexample? • Is the converse true? If no, what is a counterexample? If the polygon has 4 congruent sides, then the polygon is a square. If the polygon is a square, then the polygon has 4 congruent sides. No, counterexample: rhombus YES!!
Logical Chains • Conditionals that can be linked together are called logical chains. • Consider the following silly conditionals: If cats freak, then mice RUN. If sirens shriek, then dogs howl. If dogs howl, then cats freak. • Prove that the following conditional follows logically from the three given conditionals: If sirens shriek, then mice RUN. Hypothesis:
If-Then Transitive Property Given: You can conclude: If A then B, and if B then C If A then C. Example: Given: If there is a parade, then fireworks will go off. If it is July 4, then flags are flying. If flags are flying, then there is a parade. Prove: If it is July 4, then fireworks go off. Hypothesis:
