3. Translate the following argument into symbolic for, and determine whether it's logically correct by constructing a truth table.
-If the team wins, then the team remains in the playoffs. If the team stays in the playoffs, then the championship is within reach. Therefore, if the team wins, the championship is within reach.
To translate the argument into symbolic form, let's assign propositions to represent the statements involved in the argument:
P: The team wins
Q: The team remains in the playoffs
R: The championship is within reach
The first statement "If the team wins, then the team remains in the playoffs" can be represented as P → Q.
The second statement "If the team stays in the playoffs, then the championship is within reach" can be represented as Q → R.
The conclusion "If the team wins, the championship is within reach" can be represented as P → R.
Now, let's construct a truth table to determine whether the argument is logically correct:
| 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, there are no rows where the premise statements (P → Q and Q → R) are true, but the conclusion statement (P → R) is false. Therefore, the argument is logically correct since the conclusion always follows from the premises.
Thus, if the team wins, the championship is within reach based on the given premises.