1.7 Sub Chapter Notes

 

Forms of Conditional Statements • Logic plays a role in relating facts and drawing conclusions. • A conditional statement has the form: if p then q. • The contrapositive of a conditional is equivalent to the conditional. A conditional statement links two sentences together. It declares that the second sentence is true or false depending on whether the first sentence is true or false. In this example of a conditional statement, the fact of having an umbrella in the second sentence depends on whether it is raining or not raining as stated in the first sentence. In this version of the same example, the second sentence is stated first, followed by the first sentence stated last. Also, the opposite of each sentence is used. This use of the two clauses is the contrapositive version of the original conditional statement. It is always true or false as the original conditional statement is true or false. The converse changes the order of the two clauses. Logically, this combination cannot be held to be true without additional information. The inverse retains the same order of the clauses. However, the opposite of each clause is used. Logically, this combination cannot be held to be true without additional information. This chart summarizes all the ways two clauses can be connected with each other. Notice: Given that the conditional statement is true, only the contrapositive is also true.

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Inductive Reasoning • Use inductive reasoning to continue patterns. • The Fibonnaci sequence is found in many real-world situations and its pattern can be extended through inductive reasoning. Inductive reasoning is a method of analyzing known items to find a pattern. Then the pattern is used to predict what elements might be added to continue the pattern. In this example, the pattern seems to be that adding 1 to each term creates the succeeding term. Using that pattern, one can predict what the next elements might be if the sequence continues. Inductive reasoning concedes that more than one pattern might be available. In this example, two patterns are given. Are there more? In the first pattern, each term is seen as a multiple of 5 beginning with 5 x 1. In the second pattern, each term is analyzed based on its final digit and the fact that the first digits represent the counting numbers in order. It is worth noting that both of these patterns generate the same sequence of new elements. This is the Fibonacci sequence, probably the most famous and useful sequence mathematicians have. Always remember its first three terms and the rest of the sequence can be generated. Given the first five elements, the pattern emerges that each element is the sum of the two that precede it. Sequences involving algebraic expressions follow the same road to analysis as other sequences. In this example, the analysis reveals that each element contains two terms and the second of those terms always is +1. The first of those terms is increasing by 2x from one term to the next. From these conclusions, new elements can be predicted.

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