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poutsos.A
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a) AxB =Φ <====> [ (a,b)εAxB <-----> (a,b)εΦ] <=====>[(aεA & bεB)<------>(aεΦ & bεΦ)] <====>[(aεA----->aεΦ) & (bεB------>bεΦ)] ======>(aεA----->aεΦ)====> A= Φ ====> A=Φ v B=Φ.
adriank said:It looks like the original problem is to prove [tex]A \times B = \varnothing \Leftrightarrow A = \varnothing \vee B = \varnothing[/tex]. He is writing Φ to mean the empty set.
Hurkyl said:What are the individual steps?
Incidentally, it's clear that you've made a mistake: along the way, you asserted that
AxB = Φ ====> A = Φ
but it's easy to construct counterexamples...
That isn't valid rule of inference. Here is a truth assignment v that invalidates it:poutsos.A said:and a propositional logic law [( p&q <-----> r&s)<=====> (p---->r)&(q----->s)]
Hurkyl said:That isn't valid rule of inference. Here is a truth assignment v that invalidates it:
(I'm abbreviating true and false as T and F, respectively)
v(p) = T, v(q) = F, v(r) = F, v(s) = T
v(p&q <---> r&s) = (T&F <---> F&T) = (F <---> F) = T
v((p-->r)&(q-->s)) = (T-->F)&(F-->T) = F&T = F
Because these formulas have different truth values under the given truth assignment,
[( p&q <-----> r&s)<=====> (p---->r)&(q----->s)] is not valid in Boolean logic.
Ah, but there is: propositional logic is sound: the rules of inference in propositional logic cannot prove a result that is semantically invalid.poutsos.A said:There is no theorem to support your argument .
Intuitionistic logic (and other logics) use different rules of inference for propositional logic and for first-order logic. When referring to classical propositional and first-order logic, I attach the adjective 'Boolean' for added specificity. I'm pretty sure this is an established convention.There is no Boolean Logic but Boolean Algebra and propositional logic.
adriank said:You claim you proved that [tex]A \times B = \varnothing \Longrightarrow A = \varnothing[/tex], but that is clearly false (so the proof is invalid somewhere): take A to be any nonempty set and B to be the empty set, and you have a counterexample. What Hurkyl posted above explains where your proof went wrong.
What the heck are you talking about? That your argument asserts AxB =Φ ====> A= Φ is as plain as day.poutsos.A said:But i never proved that
Hurkyl said:What the heck are you talking about? That your argument asserts AxB =Φ ====> A= Φ is as plain as day.
Hurkyl said:Ah, but there is: propositional logic is sound: the rules of inference in propositional logic cannot prove a result that is semantically invalid.
More generally, given a set H of hypotheses and a conclusion C, we have the following theorem:
Theorem: H syntactically implies C if and only if H semantically implies COne direction of this theorem is soundness, the other completeness. This is also a theorem of (Boolean) first-order logic in general, not merely of (Boolean) propositional logic.
Intuitionistic logic (and other logics) use different rules of inference for propositional logic and for first-order logic. When referring to classical propositional and first-order logic, I attach the adjective 'Boolean' for added specificity. I'm pretty sure this is an established convention.
Incidentally, the above is irrelevant to the hole in your derivation -- the inference you tried to use is simply not one of the basic rules of inference of propositional logic. The semantic proof I gave was meant to make that fact more obvious.
AxB = Φ means that the product of A and B is equal to the empty set. This means that there are no elements in common between the sets A and B.
A=Φ v B=Φ means that either A or B (or both) is equal to the empty set. This is different from AxB = Φ, which means that the product of A and B is equal to the empty set.
If A=Φ v B=Φ is true, it means that either A or B (or both) is equal to the empty set. This can happen if one or both of the sets have no elements or if they have elements that are not in common with each other.
Yes, A and B can be any type of sets in the equation AxB = Φ. The equation only specifies that the product of A and B is equal to the empty set, but it does not limit the types of sets that A and B can be.
The equation AxB = Φ is often used in set theory and abstract algebra. It can also be used in other fields such as computer science and mathematical logic to represent the concept of an empty set or to prove certain theorems.