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2.1&2.2 Intro to Proofs & Logic

2.1&2.2 Intro to Proofs & Logic. Warm-Up: Assuming that you are paying, is it cheaper to take one friend to the movies twice, or two friends to the movies at the same time? Ms. Newgas has two coins which total 35 cents. Since one of the coins is not a dime, what are the two coins?.

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2.1&2.2 Intro to Proofs & Logic

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  1. 2.1&2.2 Intro to Proofs & Logic Warm-Up: Assuming that you are paying, is it cheaper to take one friend to the movies twice, or two friends to the movies at the same time? Ms. Newgas has two coins which total 35 cents. Since one of the coins is not a dime, what are the two coins? Objectives: -Understand the meaning of the term ‘’proof’’ -Define conditionals, hypothesis, conclusion, deduction, converse, counter example, & logical chain

  2. Proof: A convincing argument that uses logic to show that a statement is true. In mathematics a proof starts with things that are agreed on (called postulates or axioms) then logic is used to reach a conclusion. There are many different kinds of proofs.

  3. Suppose two squares are cut from opposite corners of a chessboard. Can he remaining squares be completely covered by 31 dominos? If your answer is yes, can you offer a method for showing that it is possible? If your answer is no, can you explain why it cannot be done? In either case you would be giving a proof of your answer.

  4. The altered chessboard cannot be covered by 31 dominos because of the pattern of a chessboard each domino must cover on dark square and one light square. • Any arrangement of dominoes must cover the same number of dark and light squares. The squares that were cut off are the same color, leaving more squares of one color than the other. • Therefore it is not possible to cover the altered board with 31 dominoes

  5. What if we cut off one dark square and one light square, could we cover the board with 31 dominos?

  6. http://ed.ted.com/lessons/scott-kennedy-how-to-prove-a-mathematical-theoryhttp://ed.ted.com/lessons/scott-kennedy-how-to-prove-a-mathematical-theory NIM GAME page 79 of text Play to 50 game

  7. Conditionals: A statement that can be written in the form “if p then q” where ‘p’ is the hypothesis and ‘q’ is the conclusion.

  8. Hypothesis: The phrase following the word if in the conditional statement.

  9. Conclusion: The phrase following the word then in the conditional statement; the final statement of an argument.

  10. Converse: The statement formed by interchanging the hypothesis and conclusion of a conditional statement.

  11. Counterexample: An example that proves that a statement is false.

  12. All dogs are mammals. Conditional: Hypothesis: Conclusion: Converse: Counterexample:

  13. All rectangles are parallelograms. Conditional: Hypothesis: Conclusion: Converse: Counterexample:

  14. Logical Chain: A series of logically linked conditional statements.

  15. Deduction (deductive reasoning): The process of drawing conclusions by using logical reasoning in an argument.

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