Texas A&M University-Corpus Christi / Dept. of Computing and Mathematical Sciences

MATH 2305 §002 -- Discrete Mathematics I -- Spring 2003

Chapter 1 Study Questions

Section 1.1 (Statements and Representations)

  1. Formal logic provides the foundation for the organized, careful method of thinking that characterizes any reasoned activity.
  2. If a symbol A represents a true statement and the statement B represents a false statement then the compound statement A /\ B (or A and B) has the truth value false.
  3. In the implication A -> B, A stands for the 'precedent' statements and B stands for the 'considerate' statement.
  4. The statement 'Peter is tall and thin' only has one correct negation.
  5. Negation is a unary connective.

Section 1.1 (Evaluating Well-Formed Formulas)

  1. Any combination of statement letters, connectives and parentheses is a legitimate string.
  2. Implication precedes negation in applying connectives in a wff.
  3. A truth table with 3 statement letters has 3^2 = 9 rows.
  4. A tautology is true no matter what truth values are assigned to its statement letters.
  5. There are 7 equivalences known as De Morgan's Laws.
  6. Many search engines use + to represent the connective 'and' and - to represent 'or.'
  7. Expressions can be simplified by replacing some wff's by equivalent wff's.

Section 1.2

  1. Any true implication is a valid argument.
  2. Each statement in a proof sequence is a wff.
  3. Justification for each step in a proof sequence is required.
  4. To use a derivation rule (as in Table 1.13), wwfs must exactly match the rule pattern.
  5. In the deduction method, the consequent of the conclusion of an argument is added to the list of its hypotheses.
  6. Verbal arguments have to make sense in order to be valid.

Section 1.3

  1. The predicate x > 0 can be true or false depending on the value of x.
  2. The 'for all' quantifier is the only one there is.
  3. The interpretation of a statement can be true or false.
  4. In the statement P(y) /\ P(5), y is a free variable.
  5. In the statement (Ax) ( P(x,y) /\ P(x,5) ) is in the scope of (Ax). [Here the right-side-up A signifies the universal quantifier.--Does anyone know the html code for for the upside-down A?]
  6. Negating a existential statement results in an existential statement.
  7. A predicate wff is said to be valid if it is true in all possible interpretations.

Section 1.4 (Instantiation and Generalization)

  1. Arguments can have predicate statements as hypotheses or conclusions.
  2. 'Socrates is mortal' is a predicate statement.
  3. Existential Instantiation justifies stripping off an existential quantifier.
  4. Existential instantiation can be done at any stage in a proof sequence.
  5. The wff P(x) -> (Ax) P(x) is valid.
  6. The wff P(x) -> (Ex) P(x) is valid.

Section 1.5 (Other Rules, Verbal Arguments)

  1. Application of rules in predicate logic is somewhat mechanical.
  2. In example 30, instantiation is used before generalization in the proof sequence.
  3. Hypotheses in the conclusion of a predicate argument can be used in a proof sequence.
  4. The result of a second existential instantiation in Example 34 is P(a) /\ Q(a).
  5. Propositional logic is expressive enough to capture relationships in predicate arguments.