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Mathematical Reasoning

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Marzila Mohamed
Marzila Mohamed

This document discusses different types of mathematical statements including: - Simple statements that can be true or false such as "Garfield is a cartoon character." - Quantified statements using words like "all" or "some" such as "All squares have 4 sides." - Negated statements formed by adding "not" which have the opposite truth value such as "15 + 20 does not equal 35." - Compound statements combining two simple statements with "and" or "or" such as "3 is a prime number and 3 is an odd number." Compound statements with "and" are only true if both parts are true, while statements with "or" are true if either part is true.

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Statement A statement is a mathematical sentence which can be judged to be true or false
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
Quantifiers Words that indicate number of objects or cases referred to in a statement. Example : All, Every, Any Some, Several, One of, part of
Examples True Statements All squares have 4 sides Some kites have one axis of symmetry All cows have 4 legs.
Examples False Statement All girls like pink. All fractions are smaller than one
Construct a true statement using the quantifier all or some.
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
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.
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.
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.
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.
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.
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.
Online Activity http://www.regentsprep.org/Regents/math/tables/PracAnd.htm

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Mathematical Reasoning

  • 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.