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Mathematical Reasoning
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Mathematical Reasoning

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  • 1. Statement A statement is a mathematical sentence which can be judged to be true or false
  • 2. Example
    • “Garfield is a cartoon character."
    • a true statement.
    • "A pentagon has exactly 4 sides."
    • is a false statement.
    • "Did you get that one right?"
    • Not a statement
    • “Stop the bus.”
    • Not a statement
  • 3. Quantifiers
    • Words that indicate number of objects or cases referred to in a statement.
    • Example : All, Every, Any
    • Some, Several, One of, part of
  • 4. Examples – True Statements
    • All squares have 4 sides
    • Some kites have one axis of symmetry
    • All cows have 4 legs.
  • 5. Examples – False Statement
    • All girls like pink.
    • All fractions are smaller than one
  • 6. Construct a true statement using the quantifier ‘all’ or ‘some’.
  • 7. Negation
    • a negation of a statement is formed by placing the  word "not" or no into the original statement. 
    • The negation will always have the opposite truth value of the original statement.  
    • Under negation, what was TRUE, will become FALSE - or - what was FALSE, will become TRUE
  • 8.
    • Examples:
    • 1. Original Statement:  "15 + 20 equals 35."  (is true) Negation:  "15 + 20 does not equal 35."  (is false)
    •   2. "A dog is a cat."  is a false statement. "A dog is not a cat." is a true statement.
  • 9. Compound Statement
    • A compound statement can be formed from 2 given statements using the word ‘and’ or ‘or’.
    • Examples :
    • 3 is a prime number
    • 3 is an odd number
    • 3 is a prime and odd number.
  • 10. Truth Value of a compound statement using ‘and’
    • a conjunction is a compound sentence formed by combining two statements (or facts) using the word "and."   A conjunction is true only when BOTH statements (or facts) are true.
  • 11.
    • Examples:
    • 1.  "Blue is a color and 7 + 3 = 10."  (T and T = T)      Since both facts are true, the entire sentence is true.  
    • 2.  "One hour = exactly 55 minutes and one minute = exactly 60 seconds."       (F and T = F)  Since the first fact is false, the entire sentence is false.
    • 3.  "3 + 4 = 6 and all dogs meow."  (F and F = F)      Since both facts are false, the entire sentence is false.
  • 12. Truth Value of a compound statement using ‘or’
    • a disjunction is a compound sentence formed by combining two statements (or facts) using the word "or." 
    • A disjunction is true when EITHER or BOTH sentences (or facts) are true.
  • 13.
    • Examples:
    • 1.  "Blue is a color or 7 + 3 = 10." (T or T = T)  Since both facts are true, the entire sentence is true.  
    •   2.  "One hour = exactly 55 minutes or one minute = exactly 60 seconds." (F or T = T)  Since the second fact is true, the entire sentence is true.
    •   3.  "3 + 4 = 6 or all dogs meow." (F or F = F) Since both facts are false, the entire sentence is false.  
    • 4.  "The word cat has 3 letters or the word dog has four letters."  (T  or F = T)  Since the first fact is true, the entire sentence is true.
  • 14. Online Activity
    • http://www.regentsprep.org/Regents/math/tables/PracAnd.htm