2.1 Inductive reasoning, conjectures, and counterexamples

Front

What is inductive reasoning?

Back

Inductive reasoning is a type of reasoning where conclusions are drawn based on observations and patterns.

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What is a conjecture?

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A conjecture is a statement believed to be true based on observations.

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What is a counterexample?

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A counterexample is an example that disproves a conjecture or statement.

Front

Provide a counterexample to the statement: 'If a four-sided shape has two sides the same length, then it must be a rectangle.'

Back

A square is a counterexample because it has two sides of the same length but is not a rectangle.

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Provide a counterexample to the claim: 'If two angles are supplementary, then they are not congruent.'

Back

The angles 90° and 90° are supplementary and congruent.

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Provide a counterexample to the statement: 'If two angles are supplementary, then they are both 90°.'

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The angles 100° and 80° are supplementary but not both 90°.

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What is the difference between inductive and deductive reasoning?

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Inductive reasoning involves making generalizations based on specific observations, while deductive reasoning involves drawing specific conclusions from general principles.

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What is an example of inductive reasoning?

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Observing that the sun has risen in the east every day and concluding that it will rise in the east tomorrow.

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What is the role of a counterexample in mathematics?

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A counterexample is used to show that a conjecture or statement is false.

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How can you identify a conjecture?

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A conjecture can be identified as a statement that is believed to be true but has not yet been proven.