Propositional Logic Practice
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Questions and Answers

(p ∨ ¬q)∧r

((T ∨ F) ∧ T) = T

In an implication/conditional statement 'If it rains, I will stay home', when is the sentence true?

  • When it rains and you stay home (correct)
  • When it doesn't rain
  • When it doesn't rain and you stay home
  • When it rains, regardless of staying home or not
  • In a biconditional statement 'You can take the train if and only if you buy a ticket', the statement is false if you have a ticket but the ticket checker refuses to let you board.

    True

    Define a necessary condition in logical statements.

    <p>A necessary condition is a condition that must be satisfied for the outcome to be true.</p> Signup and view all the answers

    Match the following terms with their meanings in logic:

    <p>Sufficient condition = Hypothesis is sufficient to conclude the outcome Necessary condition = Condition that must be met for the result to be true Converse = Switching the premise and conclusion of an implication statement Biconditional = Equivalent to saying 'if and only if'</p> Signup and view all the answers

    Study Notes

    Propositional Logic

    • Propositional logic deals with statements that can be either true (T) or false (F)
    • A proposition is a statement that can be either true or false

    Implication/Conditional Statement

    • If it rains, I will stay home.
    • The sentence is true if it is raining and you are staying home.
    • The sentence does not say anything about what if it does not rain, so if it does not rain, the sentence is true whether you stay home or not.
    • The sentence is false if it rains and you go out anyway.

    Hypothesis and Conclusion

    • If it rains, I will stay home.
    • The if clause is called the hypothesis/premise, and the remaining clause is called the conclusion/consequence.
    • We assume that the hypothesis is true in order to verify the validity of the conclusion.
    • The conclusion is the outcome of the hypothesis.

    Implication/Conditional Statement (continued)

    • Let P and Q be propositions.
    • P : It rains
    • Q : I will stay home
    • P→Q means "If P, then Q" or "P implies Q"
    • P is called premise/hypothesis
    • Q is called conclusion/consequence

    Sufficient Condition

    • If it rains, we know that we will find you home.
    • In other words, knowing that it is raining is sufficient to know that you are home.
    • Generally, the hypothesis is a sufficient condition for the conclusion.
    • For the proposition p→q, p is the sufficient condition for q.

    Necessary Condition

    • You will pass only if you study.
    • The sentence is true if you study and you pass.
    • If you don't study, you may not pass.
    • Note that while you may fail for any possible reason, you must study in order to pass.
    • In another word, in order to pass, it is necessary that you study.

    Necessary Condition and Sufficient Condition

    • Generally, the conclusion is a necessary condition for the hypothesis, and the hypothesis is a sufficient condition for the conclusion.
    • For p→q, q is a necessary condition for p, and p is a sufficient condition for q.

    Different Ways of Expressing Conditional Statement

    • If p then q
    • p only if q
    • p implies q
    • p is sufficient for q
    • q is necessary for p

    Biconditional

    • You can take the train if and only if you buy a ticket.
    • The sentence is true if you have a ticket and you board the train, or if you don't have a ticket and the ticket checker does not let you board.
    • The sentence is false if you have a ticket and the ticket checker still refuses to take you, or if you don't have a ticket and you board the train anyway.
    • p↔q means "p if and only if q" or "p is equivalent to q".
    • p and q are necessary and sufficient conditions for each other.

    Converse, Contrapositive, and Inverse

    • Converse: If I stay home, it is raining.
    • Inverse: If it does not rain, I will not stay home.
    • Contrapositive: If I do not stay home, it is not raining.
    • Note that only the contrapositive is equivalent to the original statement.

    Problem

    • Express the given propositions using p, q, r, and logical connectives:
      • You get an A on the final exam.
      • You do every exercise in this book.
      • You get an A in this class.

    Translating English Sentences to Logical Expressions

    • Chapter 1: 1.1, Exercise: 7, 9, 10

    Translating Logical Expressions to English

    • Chapter 1: 1.1, Exercise: 4, 5, 6, 8

    Logical Operator Precedence

    • Chapter 1: 1.1, Exercise: (no specific exercise number mentioned)

    Truth Table

    • (A→B)∧¬(A→B)

    Exercise

    • P ∧ (Q ⇔ R)

    Thank You

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    Description

    Practice exercise on propositional logic, covering topics such as logical operators and truth tables. Test your understanding of logical statements and their implications.

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