What is the weird correspondance between subset ⊂ and implication ⇒?

Question

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 ?

Answer 1

The implication (if $P$ then $Q$) is true IFF the set of assignments (propositional, first order, whatever depending on the setting) which satisfy $P$ is a subset of the set of assignments which satisfy $Q$.

Answer 2

1. using (p ⇒ q) ≡ (¬p∨q) yielded a weird Venn diagram

   Okay, this is the Venn diagram (here, white space signifies that no element is contained there) that you are referring to:

   

   This diagram says: If it belongs to set $A,$ then it also belongs to set $B.$

2. In this answer, I displayed the Euler diagram of $$\forall x \,\Big(A(x)\to B(x)\Big)$$ as this:

   

   For your example, $A$ and $B$ denote the set of cats and cunning objects, respectively, so that “Every cat is cunning.”

3. Observe that these two diagrams correspond to each other.

Answer 3

In the usual development of set theory, the expression $A\subseteq B$ is defined to mean $\forall x(x\in A\to x\in B)$.

Similarly, $A\cup B$ denotes the set of all $x$ satisfying $x\in A\lor x\in B$, and $A\cap B$ denotes the set of all $x$ satisfying $x\in A\land x\in B$. The set operations are in a sense just shorthand for the underlying logical operations.