Use the truth table to test the validity of the following argument.
If you play varsity sports, then you are dismissed from school an hour early.
You are dismissed from school an hour early.
Therefore, you are a varsity athlete.
valid
invalid
Sorry about the repost:( But I figured out the other question!:) I've just been stuck on this for a long time and I don't want to get it wrong since I'm super torn between valid and invalid. Valid since I play varsity sports, making my hour early leave valid. Or invalid since any kid could leave an hour early... please help? Thanks
Invalid
The converse is not necessarily true.
There may be lots of kids dismissed early who do not play sports at all.
Ah, the classic truth table dilemma! Let's break it down step by step, with a touch of humor to lighten the mood:
1. If you play varsity sports, then you are dismissed from school an hour early.
Let's call this statement P. To simplify things, we'll represent "You play varsity sports" as A, and "You are dismissed from school an hour early" as B. So, P can be written as "If A, then B" or A → B.
2. You are dismissed from school an hour early.
This is statement Q, and we'll represent it as B.
3. Therefore, you are a varsity athlete.
This is the conclusion, let's call it R, and we'll represent it as A.
Now, let's construct the truth table to see if the argument is valid or invalid:
| A | B | (A → B) | B | A |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | F |
In the truth table, when both P (A → B) and Q (B) are true, the conclusion R (A) should also be true for the argument to be valid. Thankfully, in this case, it is! So, the argument is valid.
In other words, if the premise "If you play varsity sports, then you are dismissed from school an hour early" is true and "You are dismissed from school an hour early" is true, then it logically follows that "You are a varsity athlete" is also true.
No need to be torn between valid and invalid anymore! You can confidently proclaim that this argument is valid, just like your skills in understanding truth tables!
To test the validity of the argument, we can create a truth table based on the given statements:
| P (Varsity Sports) | Q (Dismissed Early) | P -> Q: (P implies Q) |
|--------------------|---------------------|-----------------------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
In the truth table, T represents true and F represents false.
According to the truth table, when both premises are true (You play varsity sports and you are dismissed early), the conclusion (You are a varsity athlete) is also true.
Therefore, the argument is valid.
To test the validity of the argument, you can create a truth table with two premises and the conclusion.
Let's represent the statement "You play varsity sports" as P and the statement "You are dismissed from school an hour early" as Q. The argument can then be expressed as:
P → Q
Q
∴ P
To construct the truth table, you need to consider all possible combinations of truth values for P and Q. There are four possible combinations:
P | Q | P → Q | Q | P
T | T | T | T | T
T | F | F | F | T
F | T | T | T | F
F | F | T | F | F
In the first row, where both P and Q are true, the conditional statement P → Q is true, and Q is also true. Therefore, the conclusion P is true.
In the second row, where P is true and Q is false, the conditional statement P → Q is false, and Q is false. Therefore, the conclusion P is false.
In the third row, where P is false and Q is true, the conditional statement P → Q is true, but Q is true. Therefore, the conclusion P is false.
In the fourth row, where both P and Q are false, the conditional statement P → Q is true, but Q is false. Therefore, the conclusion P is false.
Since there is at least one row where the premises are true, but the conclusion is false (the second row), the argument is invalid.