I have a question about disjunctions.
The symbol "v" represents "or". "~" stands for "negation of"
Here are two questions I need help with:
1. When p v ~q is false, then p is ______ and q is _____.
I think "false" belongs in both spaces.
2. When ~p v q is false, then p is ______ and q is ________.
I think "false" is the answer to both of these as well.
Am I correct?
Let's analyze each question step-by-step:
1. When p v ~q is false, then p is ______ and q is _____.
To evaluate this disjunction, we need to consider the different possibilities for p and q that would make the statement false.
- If p is false, and ~q is true (since ~q is the negation of q), then p v ~q would be true since at least one of the two operands is true. Therefore, p cannot be false.
- If q is true, and ~q is false, then p v ~q would be true since at least one of the two operands is true. Therefore, q cannot be true.
From the above analysis, we can conclude that when p v ~q is false, both p and q should be false.
2. When ~p v q is false, then p is ______ and q is ________.
Similar to the previous question, let's evaluate the possibilities for p and q that would make the statement false.
- If ~p is true (meaning p is false) and q is false, then ~p v q would be true since at least one of the two operands is true. Therefore, ~p cannot be true.
- If p is true and q is true, then ~p v q would be true since at least one of the two operands is true. Therefore, p cannot be true.
From the above analysis, we can conclude that when ~p v q is false, both p and q should be false.
In conclusion, your answers are correct: both p and q should be false in both of the given scenarios.
To determine the values of p and q in each scenario, we need to look at the conditions when the given disjunctions are false. Let's break down both questions step by step:
1. When p v ~q is false:
To find the values of p and q in this case, we need to consider when the statement p v ~q is false.
- The statement p v ~q is false when both p and ~q are false.
So, the possible values for p and q in this scenario are:
p is false and q is false.
Therefore, you are correct. In this case, p is false and q is false.
2. When ~p v q is false:
Similarly, let's consider when the statement ~p v q is false.
- The statement ~p v q is false when both ~p and q are false.
So, the possible values for p and q in this case are:
p is true and q is false.
Therefore, you are incorrect in assuming that the answer is false for both p and q in this case. The correct answer is p is true and q is false.
Remember, when a disjunction is false, it means that at least one of the disjuncts is false.