Testing Logical Equivalence

  • Thread starter axellerate
  • Start date
  • #1

Main Question or Discussion Point

Hello hello, I'm not looking just for an answer per say, but am also wondering the thought process in solving problems such as the following:

Hopefully this doesn't take up too much of someones time.

Determine whether the following equations are logically equivalent:

1) (∃x)( P(x) → Q(x) )

2) (∀x)P(x) → (∃x)Q(x)
 

Answers and Replies

  • #2
22,097
3,283
Try to write the implication in terms of other connectives.
 
  • #3
nomadreid
Gold Member
1,447
142
After you have followed micromass's suggestion, the next step can make more sense to you if you contemplate an analogy (I stress that this is a way of thinking: it would not work as a formal proof)
all quantifier like a large "and",
existence quantifier like a large "or"
"and" like "intersection"
"or" like "union"
deMorgan Laws.
Formally, if you are not an intuitionist, you can try playing around with the equivalence between "[itex]\forall[/itex]x P" and "~[itex]\exists[/itex]x ~P", or between "[itex]\exists[/itex]x Q" and "~[itex]\forall[/itex]x ~Q"

(by the way, it's "per se")
 
  • #4
MLP
32
0
If you are familiar with how to determine a formula is logically true, then you can use the fact that formulas are logically equivalent just in case their biconditional is logically true. If there is an interpretation that makes the biconditional of (1) and (2) false, then they are not logically equivalent. If there is no such interpretation, then they are logically equivalent.
 

Related Threads on Testing Logical Equivalence

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
667
  • Last Post
Replies
4
Views
3K
Replies
6
Views
697
  • Last Post
Replies
5
Views
2K
Replies
2
Views
3K
Replies
3
Views
1K
Replies
2
Views
957
Replies
6
Views
2K
Replies
3
Views
3K
Top