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MATH 135 - Statements, Proofs, Implications.docx

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MATH 135
Mukto Akash

Mathematical Statements, Proofs, and Implications Mathematical Statements: A mathematical statement is a meaningful sentence that has a definite state of being true or false. Examples: Statement? Truth Value 7 > 3 Yes True 2 = π Yes False Is 5 an even No; this is a question. N/A number? χ – 3 = 2 No; this is an open N/A sentence*. Let χ be a real Yes; χ is defined. True number. If χ > 0, 2 then χ > 0 * An open sentence is a sentence which can be a statement if its variables are defined with appropriate values. Proofs: A proof establishes the truth of a given statement beyond doubt. Example: 2 Statement: For every real number χ, χ + 1 ≥ 2χ. [Proof] Let χ be a real number. Thus, (χ – 1) is also a real number. Hence, (χ – 1) ≥ 0. Then, χ - 2χ +1 ≥ 0. 2 So, χ + 1
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