- #1
nounou
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How can we write a sentence in first-order logic that says that a graph has exactly 6 edges? i.e. G= (V,E) (logically implies) iff |E|=6
they contain the same edge, since my graph has only one edge
Just to be clear those are two equivalent formulas above- I could have put an equivalence sign but I had a thing going with the colons.nounou said:Hurkyl,
to extend
Ex(Dx) & AxAy((Dx & Dy) -> x = y) : Ex(Dx & Ay(Dy -> x = y).
I'll run through it just to give you an idea of how the process might go, but I'm not sure my answer will be correct- we'll need someone else like Hurkyl to check. Check it yourself too. Okay, there are two parts to unique existence:to make it represent two edges
There exists at least one edges and there exists at most two edges: There exists exactly two edges: There exists two unique edges:
Ex(Dx) & AxAyAz((Dx & Dy & Dz) -> x = y & y=z) : (??I have no clue??).
Am I on the right track?
nounou
First-order logic is a formal system of symbolic logic that is used to represent and reason about objects and relationships between them. It is also known as first-order predicate logic or first-order quantification.
First-order logic can be used to represent graphs by defining a set of objects and predicates that describe the nodes and edges of the graph. For example, the nodes can be represented as objects and the edges can be represented as binary predicates that connect two nodes.
Using first-order logic to represent graphs allows for precise and unambiguous descriptions of complex relationships between objects. It also allows for automated reasoning and inference, making it useful for tasks such as graph analysis and knowledge representation.
First-order logic is limited in its ability to represent certain types of graphs, such as directed or cyclic graphs. It also does not have the ability to represent probabilistic relationships or uncertainty, which may be important in some applications.
First-order logic is a subset of predicate logic, which is a type of formal logic that is used to reason about relationships between objects. Other types of logic, such as modal logic and fuzzy logic, can also be used to represent graphs and have their own advantages and limitations.