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Language of Deduction

# Language of Deduction - Math 100 Expressing Deduction...

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Math 100 Expressing Deduction Martin H. Weissman With the language of sets and numbers, one can express many interesting mathematical facts. We have also learned how to use boolean operators (AND, OR, XOR, and NOT) to combine sentences into compound sentences. But to do mathematics, one needs some more “connective tissue” to express a deductive relationship. The most basic example of this is the “If... then... ” sentence structure. 1. Basic and compound deduction Many facts in basic algebra are most conveniently expressed using an “If...then...” statement. Consider these examples: If 2 x + 3 = 5, then x = 1. If x is a real number, then x 2 is non-negative. If x is an even number, then y such that 2 y = x . If | x - 3 | = 2, then ( x = 5 or x = 1). Try to complete the following to make a true “If...then...” sentence: (1) If..., then x > 1. (2) If..., then x = 1 or x = - 1. (3) If x is an even prime number, then... (4) If x 2 = x , then... The “If...then...” statement is the most basic expression of deduction . It involves a single step of deduction – one sentence follows deductively from another. We will discuss methods of deduction later – for now, we are focusing only on the language of deduction. More often than not, mathematics involves a compound deduction. One sentence deductively follows another, which follows another, etc... Here are some templates for multi-step deductions: (1) If 2 x + 3 = 5, then 2 x = 2. It follows that x = 1. (2) If | x - 3 | = 2, then x - 3 = 2 or x - 3 = - 2. Hence, x = 5 or x = 1.

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Language of Deduction - Math 100 Expressing Deduction...

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