5.9.2: 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.
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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.
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Translate compound statements between words and symbolic form.
Connective Symbol Name and
butconjunction or disjunction, inclusive or not ~ negation if , then implies conditional, implication if and only if biconditional
- Parentheses
- Negations
- Disjunctions/Conjunctions, left to right
- Conditionals
- Biconditionals
2.3 Constructing Truth Tables
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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.
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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
2.4 Truth Tables for the Conditional and Biconditional
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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