(a) translate the argument into symbolic form and (b) determine if the argument is valid or invalid. You may compare the argument to a standard form or use a truth table.

If I can get my child to preschool by 9:45AM, then I can take the 9:00AM class.
If I can take the 9AM class, then I can be done by 2PM
If I can get my child to preschool by 8:45AM then I can be done by 2PM

5 answers

p -> q

q -> r
p -> r

valid

p_>r

20 x 20 = 100

400 To convert the argument into symbolic form, we need to assign variables to different statements and then represent the logical connections between them. Let's assign variables as follows:

P: I can get my child to preschool by 9:45AM.
Q: I can take the 9:00AM class.
R: I can be done by 2PM.

Now let's write the statements in symbolic form:

Premise 1: P -> Q (If I can get my child to preschool by 9:45AM, then I can take the 9:00AM class.)
Premise 2: Q -> R (If I can take the 9AM class, then I can be done by 2PM.)
Premise 3: P -> R (If I can get my child to preschool by 8:45AM, then I can be done by 2PM.)

Now, let's analyze the validity of the argument using a truth table:

| P | Q | R | P -> Q | Q -> R | P -> R |
|-----|-----|-----|--------|--------|--------|
| T | T | T | T | T | T |
| T | T | F | T | F | F |
| T | F | T | F | T | T |
| T | F | F | F | T | F |
| F | T | T | T | T | T |
| F | T | F | T | F | T |
| F | F | T | T | T | T |
| F | F | F | T | T | T |

In the truth table, we can see that in all the cases where all three premises are true (i.e., when P, Q, and R are all true), the conclusion is also true. Therefore, the argument is valid.

To determine if the argument is valid or invalid using a standard form, we can do a logical deduction as follows:

1. Assume P.
2. Using the first premise (P -> Q), we can conclude Q.
3. Using the second premise (Q -> R), we can conclude R.
4. Therefore, if P is true, then R is also true.

Since we have deduced the conclusion R from the premises, the argument is valid.

In summary, both the truth table method and the logical deduction method show that the argument is valid.