Name the Property of Equality that justifies this statement:
x = x
Reflexive Property
Symmetric Property
Transitive Property
Substitution Property
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
I'm really not sure about the first one. Reflexive Property? And I'm very torn with the second one, I'm leaning slightly towards valid because it suggests that the person playing (me) is dismissed from school, so therefor, I'm an athlete. But I'm not sure... Please help? Thanks
symmetric property
I know it’s kinda like you could get dismissed from School an hour early for a different reason, and it would make more sense if the first statement was converse. For example if you are dismissed from school and hour early then you play varsity sports. But it’s asking to test the Validility of the argument based on The information, and if you think about it, if it asked you to draw a conclusion based on the information you would probably say that you are a varsity athlete. Therefore I believe it is valid.
For the first question, the correct answer is the Reflexive Property of Equality. The Reflexive Property states that any quantity is equal to itself. So, the statement x = x is justified by this property.
Regarding the second question, let's analyze the argument using a truth table:
| Varsity Sports | Dismissed from School Early | Varsity Athlete |
|---------------|----------------------------|-----------------|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | True |
From the truth table, we can see that the argument is invalid. We have a counterexample in the form of the third row, where a person is dismissed from school early, but they do not play varsity sports. This disproves the argument's claim that being dismissed from school early implies being a varsity athlete.
Therefore, the correct answer is invalid.
For the first question, you are correct. The Property of Equality that justifies the statement "x = x" is the Reflexive Property. The Reflexive Property states that any quantity is equal to itself.
Now let's move on to the second question about the validity of the argument. To test the validity of the argument, we can create a truth table to evaluate the logical relationships between the statements.
Let's assign P to "You play varsity sports" and Q to "You are dismissed from school an hour early." The argument can be rephrased as follows:
P -> Q
Q
Therefore, P
Now let's create a truth table for these statements:
P | Q | P -> Q
--------------
T | T | T
T | F | F
F | T | T
F | F | T
In this truth table, T stands for true and F stands for false. The "P -> Q" column represents the implication "P implies Q." We can see that the argument is invalid because there is a row in the truth table where the premises (P -> Q and Q) are true, but the conclusion (P) is false. Since the argument is invalid, the correct answer is "invalid."
Therefore, based on the truth table, the 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" is invalid.