Logic – Bell Ringer

Symbolic Logic & Propositions A proposition is a statement that is either true or false. Questions, commands, or exclamations would not be propositions, as they are neither true nor false. Examples: It is raining. The car is out of gas. I wore shorts. Non-Examples: Clean your room. Are you hot? Ouch, that hurt!

Symbolic Logic & Propositions Not a proposition. Determine if each of the following is proposition. The door is open. Do your homework. The contract is not signed. Are you hungry? Yes, I passed my test! I bought a drink.

Compound Statements A compound statement is two or more propositions joined with logic connectives. Logic Connective Name Meaning Example ∧ Conjunction and 𝑝∧𝑞 “𝑝 and 𝑞” ∨ Disjunction one or the other OR both 𝑝∨𝑞 “𝑝 or 𝑞” Exclusive Disjunction one or the other but not both p ∨ q “𝑝 or 𝑞, but not both” ¬ Negation not ¬𝑝 “not 𝑝” ⇒ Implication If, then 𝑝⇒𝑞 ”If 𝑝, then 𝑞” ⇔ Equivalence If and only if 𝑝⇔𝑞 “𝑝 if and only if 𝑞”

Truth Tables To determine if an argument is logically valid, use a truth table. A truth table shows all possible truth outcomes of a proposition or compound statement. In IB Math Studies, there is a maximum number of propositions used for truth tables.

Truth Tables In order to correctly complete the truth tables, an understanding of logic connectives is required. True is denoted by “T” and false by “F”. To begin a truth, the first columns are simply the truth combinations of the given propositions.

Truth Tables ¬𝑝 “not” – this switches the truth value of the given proposition. 𝑝∧𝑞 “and” – in order for this to be true, both truth values must be true. p ∨ q “or” – in order for this to be true, one OR the other OR both of the truth values must be true. p ∨ q “one or the other but not both” – in order for this to be true, ONLY one of these values can be true.

Implication Let p: I sell 10 houses. Let q: I get a bonus. Using the true/false truth values for p and q, can you determine when “If I sell 10 houses, then I get a bonus” will be true. The only way for an implication to be false is for “If TRUE, then FALSE”.

Equivalence Let p: I sell 10 houses. Let q: I get a bonus. Using the true/false truth values for p and q, can you determine when “I sell 10 houses, if and only if, I get a bonus” Equivalence is only true when the individual truth values are equivalent (both are true or both are false).

Truth Tables – Guided Practice Construct a truth table for the following compound propositions: 𝑝∧¬𝑞 ¬ 𝑝∨𝑞 ¬𝑞⇒𝑝 𝑝∧𝑟 ⇔(¬p ⋁ q)