Exercise 2: The Power Set of {a,b,c} and the Subset Definition — Possible Solution
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LISTING EVERY SUBSET OF A = {a, b, c}
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Working through every possible combination of elements, from smallest
to largest:

  Size 0 (nothing included):     {}
  Size 1 (one element each):     {a}, {b}, {c}
  Size 2 (two elements each):    {a, b}, {a, c}, {b, c}
  Size 3 (everything):           {a, b, c}

Full power set:
  P(A) = { {}, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c} }

CONFIRMING THE COUNT MATCHES 2^|A|
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Counting the list above: 1 (size 0) + 3 (size 1) + 3 (size 2) + 1
(size 3) = 8 total subsets. |A| = 3, so 2^|A| = 2^3 = 8 - the count
matches exactly, confirming this chapter's own formula.

THE QUANTIFIED DEFINITION OF A SUBSET B B
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  A c= B  ==  for all x, ( x in A -> x in B )

In words: for any x at all, if x happens to be an element of A, then
it must also be an element of B.

WHY -> IS THE CORRECT CONNECTIVE, NOT ^
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Using ^ instead - "for all x, (x in A ^ x in B)" - would claim that
EVERY x in the entire universe of discourse is simultaneously an
element of both A and B, which is a completely different and far
stronger (usually false) statement. It would mean A and B are
identical to each other and to the entire universe of possible
elements.

The -> version only makes a claim about elements that are actually in
A - for anything not in A, x in A is false, and per Chapter 2's own
truth table, an implication with a false premise is automatically
true regardless of the conclusion. This means elements outside A
never cause the subset statement to fail; only an element that's in A
but NOT in B could ever make "A c= B" false - which is exactly the
correct, intended meaning of "every element of A is also in B."

WHY THIS WORKS AS AN ANSWER
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It lists every subset methodically by size to avoid missing any,
confirms the resulting count against the 2^|A| formula, restates the
quantified subset definition, and explains specifically why ->
captures the intended meaning while ^ would incorrectly claim A and B
are identical to the entire universe of elements.