MIW.pdf - MATHEMATICAL LANGUAGE AND SYMBOLS “Like any language mathematics has its own syntax and rules.” Learning the Language of

MIW.pdf - MATHEMATICAL LANGUAGE AND SYMBOLS “Like any...

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MATHEMATICAL LANGUAGE AND SYMBOLS “Like any language, mathematics has its own syntax, and rules.”
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Learning the Language of Mathematics (Jamison, 2000) Unlike the language of ordinary speech, mathematical language is nontemporal devoid of emotional content precise Example: The word “any” in ordinary speech is ambiguous. Can anyone work this problem? (existential qualifier) Anyone can do it! (universal qualifier)
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Definitions (Jamison, 2000) A definition is a concise statement of the basic properties of an object or concept which unambiguously identify that object or concept. Every concept is defined as a subclass of a more general concept called genus . Each special subclass of the genus is characterized by special features called the species . Good Definition A rectangle is a quadrilateral all four of whose angles are right angles.
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Definitions (Jamison, 2000) Poor Definition (not concise) A rectangle is a parallelogram in which the diagonals have the same length and all the angles are right angles. It can be inscribed in a circle and its area is given by the product of two adjacent sides. Poor Definition (not basic) A rectangle is a parallelogram whose diagonals have equal lengths. Bad Definition (ambiguous) A rectangle is a quadrilateral with right angles. Unacceptable Definition (no genus) Rectangle: has right angles
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Important Statements 1. A universal Statements says that a certain property is true for all elements in a set. Example: All positive numbers are greater than zero. 2. A conditional statements says that if one thing is true then some other thing also has to be true. Example: If 378 is divisible by 18, then 378 is divisible by 6. 3. Given a property that may or may not be true, an Existential Statement says that there is at least one thing for which the property is true. Example: There is a prime number that is even.
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Universal Conditional Statements - - - - Universal statements contain variation of the words “for all “ and conditional statements contain versions of the words ”if then”. A universal conditional statement is a statement that is both universal and conditional Example: For all animals a , if a is a dog, then a is a mammal. Rewriting this: If a is a dog, then a is a mammal. If an animal is a dog, then the animal is a mammal. For all dog a, a is a mammal. All dogs are mammal.
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Let’s try this… a. b. c. d. e. Fill in the blanks to rewrite the following statement. For all real numbers x, if x is nonzero then x 2 is positive If a real number is nonzero, then its square _____________. For all nonzero real numbers x, ____________. If x ____________, then _____________. The square of any nonzero real number is ______________. All nonzero real numbers have ________________.
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Answer: a. b. c. d. e. For all real numbers x, if x is nonzero then x 2 is positive If a real number is nonzero, then its square is positive.
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