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:
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 ⇔?
Best Answer
Two expressions are logically equivalent if and only if their truth values are equivalent. The first two expressions you listed (P→Q)∧(Q→P), (P∨Q)∧(¬P∨¬Q) are therefore not equivalent.
The second pair of expressions, (P→Q) and ¬P∨Q however, are equal (as you correctly determined). In fact, one definition of (P→Q) is ¬P∨Q.
The double arrow means if and only if, so that if you have a statement p ⇔ q, then you need to show p → q AND q → p (which is coincidentally logically equivalent to the third column you've written). It is unnecessary for determining whether two expressions are logically equivalent though, nor can I see the relation in the example.
Reading a quick excerpt for the first book I could find under the search "double arrow logical equivalence":
"Another way to determine whether two propositions are logical equivalent is to join two propositions together using the double arrow (⇔) and then determine whether the resulting biconditional forms a tautology... thus, if a biconditional is a tautology, then the left and right sides of the biconditional are logically equivalent to each other".
Source: Symbolic Logic: Syntax, Semantics, and Proof By David W. Agler, Pg. 82, Chapter 3