[Math] Relation between implication and subset relation

logic

If $a \implies b$ can we then generally say that $a \subseteq b$ ?
For example:
if $a: x > 15$ and $b: x >10$ then clearly $a \implies b$ and if we look at the sets represented by a $\lbrace16,17,18..\rbrace$ and b $\lbrace 11,12,13… \rbrace$ it is also obvious that $a \subset b$.

Best Answer

No but you can say:

Let P be (x ∈ A)
Let Q be (x ∈ B)

A ⊆ B ≡ ∀x, P → Q


⊆ is for sets
→ is for propositions


Note: You cannot say the same for A ⊂ B
Why?

Let A be {1}
Let B be {1}
Let P be (x ∈ A)
Let Q be (x ∈ B)

A ⊂ B is false.
∀x, P → Q is true.
Therefore A ⊂ B cannot be equivalent to ∀x, P → Q

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