Subsets
Definition: The set A is a subset of the set B, denoted A ⊆ B, if every element of A is also an element of B. Symbolically, A ⊆ B if ∀x[x ∈ A → x ∈ B] is true, and conversely.
If A is a subset of B, we say that B is a superset of A, and write B ⊇ A. Clearly every set B is a subset of itself, B ⊆ B. (This is because, for any given x, x ∈ B → x ∈ B is ‘automatically’ true.) Any other subset of B is called a proper subset of B. The notation A ⊂ B is used to denote ‘A is a proper subset of B’. Thus A ⊂ B if and only if A ⊆ B and A = B.
It should also be noted that φ ⊆ A for every set A. In this case definition 3.2 is satisfied in a vacuous way—the empty set has no elements, so certainly each of them belongs to A. Alternatively, for any object x, the proposition x ∈ φ is false so the conditional (x ∈ φ) →(x ∈ A) is true.
To prove that two sets are equal, A = B, it is sufficient (and frequently very convenient) to show that each is a subset of the other, A ⊆ B and B ⊆ A. Essentially, this follows from the following logical equivalence of compound propositions:
(P ↔ Q) ≡ [(P → Q) ∧ (Q → P)].
We know that A = B if ∀x(x ∈ A ↔ x ∈ B) is a true proposition. We noted that to prove that a biconditional P ↔ Q is true, it is sufficient to prove both conditionals P → Q and Q → P are true. It follows that to prove
∀x(x ∈ A ↔ x ∈ B) it is sufficient to prove both ∀x(x ∈ A → x ∈ B) and ∀x(x ∈ B → x ∈ A). But ∀x(x ∈ A → x ∈ B) is true precisely when A ⊆ B and similarly ∀x(x ∈ B → x ∈ A) is true precisely when B ⊆ A. In summary:
