Propositional Logic Practice

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