Show that (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐) are logically equivalent
Question
Show that (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐) are logically equivalent
Solution 1
To show that (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐) are logically equivalent, we can use truth tables.
Step 1: Define the truth values for p, q, and r.
| p | q | r |
|---|---|---|
| T | T | T |
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
Step 2: Calculate the truth values for (๐ โ ๐), (๐ โ ๐), and (๐ โจ ๐).
| p | q | r | (๐ โ ๐) | (๐ โ ๐) | (๐ โจ ๐) |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | T | F | T | F | T |
| T | F | T | F | T | T |
| T | F | F | F | F | F |
| F | T | T | T | T | T |
| F | T | F | T | T | T |
| F | F | T | T | T | T |
| F | F | F | T | T | F |
Step 3: Calculate the truth values for (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐).
| p | q | r | (๐ โ ๐) | (๐ โ ๐) | (๐ โจ ๐) | (๐ โ ๐) โจ (๐ โ ๐) | ๐ โ (๐ โจ ๐) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | F | T | F | T | T | T |
| T | F | T | F | T | T | T | T |
| T | F | F | F | F | F | F | F |
| F | T | T | T | T | T | T | T |
| F | T | F | T | T | T | T | T |
| F | F | T | T | T | T | T | T |
| F | F | F | T | T | F | T | T |
As you can see, the truth values for (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐) are the same for all possible values of p, q, and r. Therefore, (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐) are logically equivalent.
Solution 2
To show that (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐) are logically equivalent, we can use truth tables.
Step 1: Define the truth values for p, q, and r.
| p | q | r |
|---|---|---|
| T | T | T |
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
Step 2: Calculate the truth values for (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐).
Remember that the implication p โ q is false only when p is true and q is false, otherwise it's true. The disjunction p โจ q is true if either p or q is true.
| p | q | r | (๐ โ ๐) | (๐ โ ๐) | (๐ โ ๐) โจ (๐ โ ๐) | ๐ โจ ๐ | ๐ โ (๐ โจ ๐) |
|---|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T | T |
| T | T | F | T | F | T | T | T |
| T | F | T | F | T | T | T | T |
| T | F | F | F | F | F | F | F |
| F | T | T | T | T | T | T | T |
| F | T | F | T | T | T | T | T |
| F | F | T | T | T | T | T | T |
| F | F | F | T | T | T | F | T |
Step 3: Compare the columns for (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐).
As you can see, the truth values in these columns are the same for all possible values of p, q, and r. Therefore, (๐ โ ๐) โจ (๐ โ ๐) and ๐ โ (๐ โจ ๐) are logically equivalent.
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