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This text explores essential concepts in inductive and deductive reasoning, focusing on conjectures, counterexamples, conditional statements, negations, and logical relationships. A conjecture is defined as an unproven statement based on observations, while counterexamples serve to disprove such conjectures. The text outlines how to form conditional statements with hypotheses and conclusions, and how to derive inverse, converse, and contrapositive statements. It concludes with exercises to reinforce understanding of these principles.
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Vocabulary A conjecture is an unproven statement that is based on observations. You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case.
Write a conjecture • Look at the patterns below and write a conjecture for the next number in the sequence.
Vocabulary • A counterexample is a specific case for which the conjecture is false.
Counterexample Find a counter example to show that the following conjecture is false. The sum of two numbers is always greater than the larger number.
Vocabulary A conditional statement is logical statement that has two parts, a hypothesis and a conclusion. The hypothesis is the “if” part of the conditional statement. The conclusion is the “then” part of the conditional statement.
Writing a conditional statement: The hypothesis tells you what you are talking about, and the conclusion describes the hypothesis.
Writing a conditional statement • Writing the following statements in if-then form. 1. Two angles that make a linear pair are supplementary. 2. All 90o angles are right angles.
Vocabulary • The negation of a statement is the opposite of the original.
Negation • Negate the following statements. 1. The ball is red. 2. The cat is not black.
Vocabulary • The inverse of a conditional statement negates the hypothesis and conclusion • The converse of a conditional statement switches the hypothesis and conclusion. • The contrapositive of a conditional statement takes the inverse of the converse.
Writing statements • Write the inverse, converse, and contrapositive of the conditional statement: “If two angles form a linear pair, then they are supplementary.” Which statements are always true?
Vocabulary • If a conditional statement and its converse are both true, then we can write it as a biconditional statement by using the phrase if and only if instead of putting it in if-then form. 1. If an angle has 90˚, then it is a right angle.
Conclusion • Explain how to write a conditional statement, its inverse, converse, and contrapositive.
Assignment Pg. 75 #5-17 odd Pg. 82 #4-14 even, 19, 21
