TMUA Logic and proof

• What is the fundamental notion at the heart of mathematical logic?

  A statement and the relationship between statements
• What can we say about statements in mathematics?

  They must be either true or false, not both
• What is the definition of a statement in this context?

  A sentence that is definitely true or false
• What is the law of the excluded middle?

  A statement can only be true or false
• Why is it acceptable if we cannot determine a statement's truth value?

  It must still be either true or false
• Give an example of a statement.

  It rained yesterday in Auckland
• Why is "The only barber in a town shaves each and every man who does not shave himself" not a statement?

  It is neither true nor false
• What is an example of a true statement?

  If $x =$$3, then x^2 =$$9$
• What is an example of a false statement?

  If $x =$$3, then x^2 =$$4$
• What is the truth value of the statement "The sum of two odd numbers is an even number"?

  True
• What are the three types of basic statements discussed?

  • Obviously true or false statements
  • Statements needing work to determine truth
  • Quantified combinations of expressions
• What does the truth value of a statement refer to?

  Whether the statement is true or false
• What does it mean for two statements to be logically equivalent?

  They have the same truth values
• Give an example of two logically equivalent statements.

  Today is Tuesday and Today is the day after Monday
• What is the process of making new statements in mathematics?

  • Combine existing statements
  • Analyze truth or falsity of combinations
  • Use logical rules to build new statements
• What does the term "not" do to a statement?

  It negates the truth value of the statement
• What is the negation of the statement "29 is a prime number"?

  29 is not a prime number
• What is the general property of the negation of a statement?

  It changes true to false and vice versa
• How does a truth table represent the relationship between a statement and its negation?

  It shows true becomes false and vice versa
• What are the two ways to display the relationship between a statement and its negation?

  • Truth table
  • Venn diagrams
• Why are statements important in mathematical logic?

  They form the basis for logical reasoning
• What distinguishes a statement from a non-statement?

  A statement can be true or false
• Why is quantification significant in statements?

  It clarifies the range of values for variables
• What is the significance of compound statements in logic?

  They show relationships between multiple statements
• What does the rule mentioned do to false statements?

  It changes them to true ones
• What does the law of the excluded middle state?

  Statements are always either true or false
• What are the two ways to display how not works for general statements?

  • Truth table
  • Diagrams (Venn diagrams)
• What does T stand for in the truth table?

  True
• What does F stand for in the truth table?

  False
• What does the first line of the truth table indicate?

  When A is true, not A is false
• What does the second line of the truth table indicate?

  When A is false, not A is true
• How do Venn diagrams represent statements A and not A?

  • Area inside A circle: A is true
  • Area outside A circle: not A is true
• How can A be thought of in set theory terms?

  A represents the set where A is true
• How can not A be thought of in set theory terms?

  As the complement of the set A
• What is the logical term used to combine statements A and B?

  And
• When is the compound statement A and B true?

  If both A and B are true
• When is the compound statement A and B false?

  If at least one of A or B is false
• What does the truth table for A and B look like?

  • A: T, B: T → A and B: T
  • A: T, B: F → A and B: F
  • A: F, B: T → A and B: F
  • A: F, B: F → A and B: F
• What does the statement "the monarch is a woman and the Prince of Wales is called Charles" illustrate?

  Both parts are true, making the statement true
• How can A and B be represented in set theory terms?

  • A ∩ B (A intersect B)
  • Both A and B must occur