ABSTRACT
A subset T of S is a set such that for all elements x of T also x is an element of S. Write T ⊂ S or S ⊃ T . A subset of S is proper if it is neither S itself nor φ. The union of a set F of sets is ⋃
S∈F S = {x : x ∈ S for some S ∈ F}
The intersection is ⋂ S∈F
S = {x : x ∈ S for all S ∈ F}
We make an exception in the case of intersections over F for F = φ, since the defining condition would be vacuous, and (supposedly) every set would be an element of that intersection, which is not viable. The union and intersection of a finite number of sets can also be written, respectively, as
S1 ∪ . . . ∪ Sn
S1 ∩ . . . ∩ Sn
Proto-definition: The ordered pair construct (x, y) with first component x and second component y should have the property that
(x, y) = (z, w)⇐⇒ x = z and y = w
14.1.3 Remark: As sets, taking (x, y) = {x, y} fails, since the elements of a set are not ordered. Taking (x, y) = {x, {y}} fails, since it may be that x = {y}. 14.1.4 Proposition: We can construct ordered pairs as sets by defining
(x, y) = {{x}, {x, y}}
Proof: We must prove that (x, y) = (z, w) if and only if the respective components are equal. One direction of the implication is clear. For the other implication, from
{{x}, {x, y}} = {{z}, {z, w}}
{x} is either {z} or {z, w}, and {x, y} is either {z} or {z, w}. Treat cases, using the Extension Principle. ///
For finite n, define recursively ordered n-tuples by
(x1, . . . , xn−1, xn) = ((x1, . . . , xn−1), xn)
14.1.5 Remark: Subsequently we ignore the internal details of the construction of ordered pair, and only use its properties. This is a typical ruse.
