Premise: "The sky is blue." (i.e., P is true)
Inference: "The sky is blue or the grass is green." (i.e., P ∨ Q)
Explanation: Since P ("The sky is blue") is true, the disjunction P ∨ Q ("The sky is blue or the grass is green") is true regardless of whether the grass is green. Only one part of the disjunction needs to be true for the entire statement to be true.
Premise: "I passed the exam." (i.e., P is true)
Inference: "I passed the exam or I missed the class." (i.e., P ∨ Q)
Explanation: Since you passed the exam, it doesn’t matter whether or not you missed the class—the statement "I passed the exam or I missed the class" will still be true because P is true.
Premise: "The lights are on." (i.e., P is true)
Inference: "The lights are on or the door is locked." (i.e., P ∨ Q)
Explanation: Since "The lights are on" is true, the disjunction "The lights are on or the door is locked" is true regardless of the door’s status. This is valid because the first part of the disjunction is true.
Premise: "The water is cold." (i.e., P is true)
Inference: "The water is cold or the air is warm." (i.e., P ∨ Q)
Explanation: The statement "The water is cold or the air is warm" is true because P ("The water is cold") is true. The truth of Q doesn’t affect the outcome since P is already true.
Premise: "The book is on the table." (i.e., P is true)
Inference: "The book is on the table or the pen is on the floor." (i.e., P ∨ Q)
Explanation: The disjunction "The book is on the table or the pen is on the floor" is true because P ("The book is on the table") is true, regardless of whether the pen is on the floor or not.
Premise: "It is snowing." (i.e., P is true)
Inference: "It is snowing or it is sunny." (i.e., P ∨ Q)
Explanation: Even if it’s not sunny, the statement "It is snowing or it is sunny" is true because P ("It is snowing") is true. In a disjunction, only one part needs to be true.
Premise: "I have my keys." (i.e., P is true)
Inference: "I have my keys or I lost my wallet." (i.e., P ∨ Q)
Explanation: Even if you didn’t lose your wallet, the statement "I have my keys or I lost my wallet" is true because P ("I have my keys") is true. The disjunction holds true as long as one component is true.
Premise: "The cat is sleeping." (i.e., P is true)
Inference: "The cat is sleeping or the dog is barking." (i.e., P ∨ Q)
Explanation: Since P is true ("The cat is sleeping"), the disjunction "The cat is sleeping or the dog is barking" is true, regardless of whether the dog is barking or not.
Premise: "The car is parked." (i.e., P is true)
Inference: "The car is parked or the engine is running." (i.e., P ∨ Q)
Explanation: Even if the engine isn’t running, the statement "The car is parked or the engine is running" is true because P ("The car is parked") is true. Only one part of the disjunction needs to be true.
Premise: "I am hungry." (i.e., P is true)
Inference: "I am hungry or I am tired." (i.e., P ∨ Q)
Explanation: Even if you aren’t tired, the statement "I am hungry or I am tired" is true because P ("I am hungry") is true. As long as one part of the disjunction is true, the whole statement holds.