What's the difference between = and <=> in logical statements?

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The discussion clarifies the distinction between the "=" sign, which indicates equality between two mathematical objects, and "<=>", which signifies logical equivalence between statements. The "=" sign asserts that two values are identical, while "<=>" indicates that two statements imply each other, thus being equivalent but not necessarily equal. The conversation highlights that logical statements can be equivalent without being equal, as seen in examples from vector spaces and mathematical structures. It emphasizes that the biconditional "<=>" cannot replace "=" since it pertains to the truth values of statements rather than numerical equality. Overall, the key takeaway is that equality and equivalence serve different purposes in mathematics and logic.
AlbertEinstein
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Difference between "=" ,<=>

It may be a little funny but what's the difference between the signs
a) equals to
b) implies and is implied by?
 
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One means equal, the other doesn't. One means that two logical statements are equivalent. The other doesn't. Is that sufficient?
 
I didn't understand. Suppose S=>T and T=>S then S <=> T does this not tell that both imply each other and hence must be equivalent.
 
But not equal. Two equivalent logical propositions are not said to be equal. Similiary 9=3^2, but you wouldn't say 3^2 if and only if 9, would you?
 
Do you mean to say that "equals to " is used in the case of numbers and "implies and is implied by " is used in logical statements?
 
I'm saying that equivalent is used in the case of logical statements, and equals is not. I am not saying where equals is used. Two mathematical objects can frequently be equivalent (or sometimes isomorphic) without actually being equal. In your other thread on residue classes, picking different choices of residue systems will give equivalent/isomorphic rings, but they are certainly not equal. If you know about vector spaces, then any finite dimensional vector space is isomorphic to its dual, but it is not equal to its dual. The vector space area is the best one for examples like this. Any two spaces of dimension n are isomorphic ((over the same field, obviously) but they are almost never going to be equal. As it is I have _never_ heard anyone refer to two logical statements being equal.
 
Most of the things went above my head. However, how two things are defined to be equal.? and what can these things may comprise of(in terms of mathematical objects)
 
Actually, ""equals to " is used in the case of numbers and "implies and is implied by " is used in logical statements" is not a bad, if slightly simplistic way to look at it. "Equals" as used in mathematics means "are the same thing". There are many, many different definitions of "equivalent" for various mathematical or logical terms. It is perfectly reasonable to say that if one statement implies and is implied by another statement, then they are "equivalent". Saying that "statement A" equals "statement B" means that they are, in fact, references to the same statement.
 
x = y means that x eequals y for any value for x and y. x <=> y means that the value of x is undefinite with respect to y, x may be equal,greatr than or less than y.
 
  • #10
gautamaish said:
x = y means that x eequals y for any value for x and y. x <=> y means that the value of x is undefinite with respect to y, x may be equal,greatr than or less than y.

I think you're misinterpreting the OP. "<=>" is supposed to mean a double arrow, and not a combination of the signs <, =, and >.
 
  • #11
We're talking about statements in logic, not numbers.

Although, I'm still wondering about
x = y means that x equals y for any value for x and y
So if x= 3 and y= 5, they are still equal?:smile:
 
  • #12
HallsofIvy said:
We're talking about statements in logic, not numbers.

Although, I'm still wondering about

So if x= 3 and y= 5, they are still equal?:smile:

no , i was not telling that, i meant that x = y means that if x is any no. y will surely be of the same value.
 
  • #13
I understood that, that's why I put the ":smile: ". Of course, the phrase "for all x and y" doesn't really apply here.
 
  • #14
Two objects a and b are equal iff for any predicate P, Pa <=> Pb. I guess this depends on what your predicates can be. 2^2 is not equal to 4 in absolutely _every_ way--for example one of them appears earlier in the sentence than the other. But I guess that wouldn't be an allowable predicate.
 
  • #15
The biconditional <=> is used in the context of logical statements (as mentioned previously) and is a relation between the truth value of those statements. But it can't take the place of = because it just doesn't make any sense. For example, given that x = y, you can't say that x <=> y because both of the terms by themself don't have a truth value. Things like "x" or "5" or "2^2" by themselves mean nothing other than just symbols.
 

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