2.8.1: Key Concepts

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    Key Concepts

    2.1 Statements and Quantifiers

    • Logical statements have the form of a complete sentence and make claims that can be identified as true or false.
    • Logical statements are represented symbolically using a lowercase letter.
    • The negation of a logical statement has the opposite truth value of the original statement.
    • Be able to
      • Determine whether a sentence represents a logical statement.
      • Write and translate logical statements between words and symbols.
      • Negate logical statements, including logical statements containing quantifiers of all, some, and none.

    2.2 Compound Statements

    • Logical connectives are used to form compound logical statements by using words such as and, or, and if …, then.
    • A conjunction is a compound logical statement formed by combining two statements with the words “and” or “but.” If the two independent clauses are represented by and , respectively, then the conjunction is written symbolically as . For the conjunction to be true, both and must be true.
    • A disjunction joins two logical statements with the or connective. In, logic or is inclusive. For an or statement to be true at least one statement must be true, but both may also be true.
    • A conditional statement has the form if , then , where and are logical statements. The only time the conditional statement is false is when is true, and is false.
    • The biconditional statement is formed using the connective for the biconditional statement to be true, the true values of and , must match. If is true then must be true, if is false, then must be false.
    • Translate compound statements between words and symbolic form.
      Connective Symbol Name
      and
      but      		 conjunction
      or disjunction, inclusive or
      not ~ negation
      if , then implies conditional, implication
      if and only if biconditional
  1. The dominance of connectives explains the order in which compound logical statements containing multiple connectives should be interpreted.
  2. The dominance of connectives should be applied in the following order
    • Parentheses
    • Negations
    • Disjunctions/Conjunctions, left to right
    • Conditionals
    • Biconditionals
    A table with four columns shows Dominance, Connective, Symbol, and Evaluate. The dominance column on the table shows a downward vertical arrow from least dominant to most dominant. The connective column on the table shows Parentheses, Negation, Disjunction or Conjunction, Conditional, and Biconditional. The Symbol column on the table shows an open bracket and a closed bracket, equivalent, an upward circumflex and a downward circumflex, a right side arrow, and a double-sided arrow. The Evaluate column on the table shows First, a downward arrow, Left to right or add parentheses to specify order because or slash and have equal dominance. a downward arrow, and last.
    Figure 2.18

    3. 2.3 Constructing Truth Tables

    • Determine the true values of logical statements involving negations, conjunctions, and disjunctions.
      • The negation of a logical statement has the opposite true value of the original statement.
      • A conjunction is true when both and are true, otherwise it is false.
      • A disjunction is false when both and are false, otherwise it is true.
    • Know how to construct a truth table involving negations, conjunctions, and disjunctions and apply the dominance of connectives to determine the truth value of a compound logical statement containing, negations, conjunctions, and disjunctions.
      Negation Conjunction (AND) Disjunction (OR)
      T F T T T T T T
      F T T F F T F T
          F T F F T T
          F F F F F F
  3. A logical statement is valid if it is always true. Know how to construct a truth table for a compound statement and use it to determine the validity of compound statements involving negations, conjunctions, and disjunctions.

    4. 2.4 Truth Tables for the Conditional and Biconditional

    • The conditional statement, if then , is like a contract. The only time it is false is when the contract has been broken. That is, when is true, and is false.
      Conditional
      T T T
      T F F
      F T T
      F F T

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