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

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 ?

• Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Mar 28 at 7:56
• You should refrain to use the term "weird" so many times (especialy for things that doesn't deserve this adjective). Mar 28 at 8:17
• No "weird" at all. There is a link connecting sets and their operations ($\cap, \cup$) with the corresponding "properties" i.e. $C = \{ x \mid C(x) \}$. Thus we have the correspondence that you highlighted between $P \cap Q$ and $\{ x \mid P(x) \land Q(x) \}$. Mar 28 at 8:24
• Wrt the "similarity" between $\subseteq$ and $\to$ we have to note that there is a difference between $\cap, \cup$ and $\subseteq$: the first two are operators producing a set from a pair of sets, while $\subseteq$ "produces" a statement involving two sets: $P \subseteq Q$ is defined as: $\forall x (P(x) \to Q(x))$, Mar 28 at 8:26
• See also math.stackexchange.com/questions/3316802/… (and tangentially math.stackexchange.com/questions/2560267/… as well). Mar 28 at 17:22

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 $$Q$$.

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.

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.