Truth table: is (P→Q)∧(Q→P) logically equivalent to (P∨Q)∧(¬P∨¬Q) ?

[salik]

Truth table: is (P→Q)∧(Q→P) logically equivalent to (P∨Q)∧(¬P∨¬Q) ?

Hi all, I am a bit confused about this truth table that I have copied down during my lessons.
The question is Use a truth table to determine whether the statement (P→Q)∧(Q→P) is logically equivalent to (P∨Q)∧(¬P∨¬Q)
This is the truth table that I have written down:

P	Q	(P→Q)∧(Q→P)	⇔	(P∨Q)∧(¬P∨¬Q)
0	0	1	0	0
0	1	0	0	1
1	0	0	0	1
1	1	1	0	1

What I am confused now is if the fourth column is it really needed (⇔) for this question?
While compiling my notes, there is another similar question where it asks if (P→Q) is logically equivalent to ¬P∨Q and that particular column (⇔) is not shown in the truth table, I assume that is because the result of (P→Q) and ¬P∨Q is the same, hence enough said?

And so, is that column needed? Otherwise, under what sort of scenario or how are the questions being phrased that would requires me to use ⇔?

 

[pka]

Hi all, I am a bit confused about this truth table that I have copied down during my lessons.
The question is Use a truth table to determine whether the statement (P→Q)∧(Q→P) is logically equivalent to (P∨Q)∧(¬P∨¬Q)
This is the truth table that I have written down:

P	Q	(P→Q)∧(Q→P)	⇔	(P∨Q)∧(¬P∨¬Q)
0	0	1	0	0
0	1	0	0	1
1	0	0	0	1
1	1	1	0	1

What I am confused now is if the fourth column is it really needed (⇔) for this question?
While compiling my notes, there is another similar question where it asks if (P→Q) is logically equivalent to ¬P∨Q and that particular column (⇔) is not shown in the truth table, I assume that is because the result of (P→Q) and ¬P∨Q is the same.

Without the truth table, how do you know that? This simple fact is we don't. That is the entire point of this exercise.
If the \(\displaystyle \iff\) column consists entirely of \(\displaystyle 1's\) then the two statements have the exact same truth-value for any combination of truth-assignments to the same individual variables. That is, two statements are logically equivalent iff they have the same truth-value for the same truth-assignments.