As i am learning mathematical logic , I noticed similarities between the concepts of logical connecters and set operators.Most of these made sense like how p ∧ q corresponds to P ∩ Q. Statement p ∧ q is true only when both p,q are true & similarly an element belongs to the set P ∩ Q only when it belongs to both P,Q.
But the one that puzzled me was implication. It did'nt really have an obvious counterpart in set theory and using (p ⇒ q) ≡ (¬p∨q) yielded a weird Venn diagram to say the least.Today I came across a problem in which I was asked to represent the statement "All cats are cunning" in formal notation which I worked out to be (∀x)[Cat(x)⇒Cunning(x)]."Now this felt very similar to how I used to use subsets to express the same idea as Ca⊂Cu:Ca={set of all cats},Cu={set of all cunning animals} during my school days.Not just this ,there are many instances where I could transform the concept of subset in quantifiers into the concept of implication.eg (∀x∈{3,4,5,...})P(x) ≡ (∀x∈N)[(x>2)⇒P(x)]
What is this weird relationship between implication and subset ?