Logical equivalence of statements with truth tables

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In summary, the two statements are not logically equivalent because they have different end results and do not have the same truth values for all combinations of S and T.
  • #1
icantadd
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Homework Statement



[tex](S \wedge T) \wedge \neg (S \wedge T)[/tex]

[tex] (S \vee T) \Rightarrow (S \wedge T) [/tex]


Are the two (predicates?) logically equivalent?

Homework Equations



Not sure, but I believe that logical equivalence means that two predicates give the same output on the same input.

The Attempt at a Solution



Worked truth tables. The end result of both where statement T is t t f f and statement S is t f t f, in both cases is t f f t. I guess I am not really sure if that means that the two predicates are logically equivalent or not. I was actually just playing with truth tables and got this on accident, and wasn't sure if the fact that they both have the same end result is worth noting or not.
 
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  • #2
icantadd said:
Not sure, but I believe that logical equivalence means that two predicates give the same output on the same input.

Correct.

I will take a look at this in the afternoon if other PF'ers haven't got to it before me...its 2.30 in the morning.
 
  • #3
icantadd said:

Homework Statement



[tex](S \wedge T) \wedge \neg (S \wedge T)[/tex]

[tex] (S \vee T) \Rightarrow (S \wedge T) [/tex]


Are the two (predicates?) logically equivalent?

Homework Equations



Not sure, but I believe that logical equivalence means that two predicates give the same output on the same input.

The Attempt at a Solution



Worked truth tables. The end result of both where statement T is t t f f and statement S is t f t f, in both cases is t f f t. I guess I am not really sure if that means that the two predicates are logically equivalent or not. I was actually just playing with truth tables and got this on accident, and wasn't sure if the fact that they both have the same end result is worth noting or not.

The way I see it is you just work the logic tables look at all possibilities for (S,T) i.e. (True, True), (True, False), (False, True), (False, False). If that's what you did and got the same result then I would believe they are logically equivalent.
 
  • #4
icantadd said:

Homework Statement



[tex](S \wedge T) \wedge \neg (S \wedge T)[/tex]

[tex] (S \vee T) \Rightarrow (S \wedge T) [/tex]


Are the two (predicates?) logically equivalent?

Homework Equations



Not sure, but I believe that logical equivalence means that two predicates give the same output on the same input.
Yes, that is correct. If this is homework, surely you could look that up in your textbook?

The Attempt at a Solution



Worked truth tables. The end result of both where statement T is t t f f and statement S is t f t f, in both cases is t f f t. I guess I am not really sure if that means that the two predicates are logically equivalent or not. I was actually just playing with truth tables and got this on accident, and wasn't sure if the fact that they both have the same end result is worth noting or not.
You just said " logical equivalence means that two predicates give the same output on the same input"
 
  • #5
First, note that

[tex]
(S \wedge T) \wedge \neg (S \wedge T)
[/tex]

will always be false (for every combination of S and T) since [tex] (S \wedge T) [/tex] and [tex] \neg (S \wedge T) [/tex] always have opposite truth values.

However, if both S and T are true, then so is

[tex]
(S \vee T) \Rightarrow (S \wedge T)
[/tex]

These two statements are not equivalent.
 

1. What is logical equivalence?

Logical equivalence is a relationship between two statements where they have the same truth value in every possible scenario. This means that if one statement is true, the other statement will also be true, and if one statement is false, the other statement will also be false.

2. How do you determine logical equivalence using truth tables?

To determine logical equivalence using truth tables, you must first list all the possible combinations of truth values for the variables in each statement. Then, you can compare the truth values of the two statements for each combination. If the truth values match for every combination, the statements are logically equivalent.

3. What is the purpose of using truth tables in determining logical equivalence?

Truth tables are a useful tool for determining logical equivalence because they provide a systematic way to list and compare all possible truth values for the variables in a statement. This helps to ensure that all possible scenarios are considered when determining logical equivalence.

4. Can two statements with different wording be logically equivalent?

Yes, two statements with different wording can be logically equivalent. Logical equivalence is based on the truth values of the statements, not the wording. As long as the truth values match for every possible scenario, the statements are considered logically equivalent.

5. How is logical equivalence different from logical implication?

Logical equivalence is a relationship between two statements where they have the same truth value in every possible scenario. On the other hand, logical implication is a relationship between two statements where the truth of one statement guarantees the truth of the other statement. In other words, if the first statement is true, then the second statement must also be true, but the reverse is not necessarily true.

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