use a truth table to test the validity of the following argument if you are on the student council then you must maintain at least a B average you are not on the student council therefore you do not maintain a B average
1. valid
2. invalid
To test the validity of the argument, we can construct a truth table:
Consider the following propositions:
A: "You are on the student council."
B: "You maintain at least a B average."
The argument can be represented as:
If A, then B.
Not A.
Therefore, not B.
Based on this representation, we can construct the truth table:
| A | B | If A, then B | Not A | Therefore, not B |
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| T | T | T | F | F |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | T | T |
We can see that there is at least one row where both premises are true (top row), but the conclusion is false. Therefore, the argument is invalid.
To test the validity of the argument, we can create a truth table representing all possible combinations of truth values for the statements involved.
Let's assign the following variables to the statements:
P: You are on the student council.
Q: You maintain at least a B average.
The argument can be rewritten as:
If P, then Q. (Premise)
Not P. (Premise)
Therefore, not Q. (Conclusion)
Using these premises, we can construct a truth table to evaluate the argument:
| P | Q | If P, then Q | Not P | Not Q |
|---|---|--------------|-------|-------|
| T | T | T | F | F |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | T | T |
In the truth table, when both premises are true (P and Not P are both T), the conclusion is also true (Not Q is T). However, in the second row, both premises are true, but the conclusion is false. Therefore, the argument is invalid, as there is a counterexample where all the premises are true, but the conclusion is false.
To test the validity of the given argument using a truth table, we need to assign truth values to each statement and examine the resulting truth values for the conclusion. Let's consider the following variable assignments:
P: You are on the student council.
Q: You maintain at least a B average.
The argument can be represented as:
Premise 1: If P, then Q.
Premise 2: ¬P.
Conclusion: ¬Q.
Let's construct the truth table:
| P | Q | ¬P | If P, then Q | ¬Q |
|---|---|----|--------------|----|
| T | T | F | T | F |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | T | T |
In the truth table, "T" represents true and "F" represents false.
The truth table reveals that in cases where P is true and Q is false, the premises are satisfied (¬P is also true), but the conclusion (¬Q) is false. Hence, the argument is invalid.
Therefore, the correct answer is:
2. invalid