Geometry: Inductive and Deductive Reasoning

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What is inductive reasoning?

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Inductive reasoning involves making generalizations based on specific observations or examples. For instance, observing that all observed houses on South Street have roofs that are falling apart leads to the conclusion that Sherry's roof is also falling apart.

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What is deductive reasoning?

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Deductive reasoning involves starting with a general statement or hypothesis and examining the possibilities to reach a specific, logical conclusion. For example, if 'All humans are mortal' and 'Socrates is a human', then 'Socrates is mortal'.

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What is a counterexample?

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A counterexample is an example that disproves a statement or proposition. For instance, the statement 'If an angle is obtuse, then it is 125 degrees' can be disproven by providing an angle like 160 degrees, which is obtuse but not 125 degrees.

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What is a sequence in mathematics?

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A sequence is an ordered list of numbers that follow a particular pattern or rule. For example, the sequence A, D, G, J follows a pattern where each letter is 3 positions forward in the alphabet.

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What is the conclusion in a logical argument?

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The conclusion is the statement that follows logically from the premises. For example, from the premises 'If it is Friday, then I will get paid' and 'It is Friday', the conclusion is 'I will get paid'.

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Define the term 'premise' in logic.

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A premise is a statement or proposition that provides the foundation for an argument. It is an assumption that leads to a conclusion.

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What is the difference between a valid and an invalid argument?

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A valid argument is one where if the premises are true, the conclusion must also be true. An invalid argument has premises that do not guarantee the truth of the conclusion.

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What is a mathematical pattern?

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A mathematical pattern is a repeated or predictable arrangement of numbers, shapes, or other mathematical objects. For example, the pattern of even numbers: 2, 4, 6, 8, ...

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What is the significance of the Law of Syllogism in deductive reasoning?

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The Law of Syllogism states that if 'If p, then q' and 'If q, then r' are both true, then 'If p, then r' is also true. It allows for drawing conclusions from multiple premises.

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What is an example of inductive reasoning?

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An example of inductive reasoning is observing that the sun has risen in the east every day of your life and concluding that the sun will rise in the east tomorrow.