Math 330: Number Systems
Advice

• Proposition 2.4'. If a < b and b ≤ c, then a < c.
• Proposition 2.4''. If a ≤ b and b < c, then a < c.
• Proposition 2.4'''. If a ≤ b and b ≤ c, then a ≤ c.

• The basic structure of a statement with a quantifier is that it has two parts ("segments").
  1. The first part is the quantification segment (or "range segment"). It states (a) the range of a variable, i.e., the set of possible values—for instance, "∀ x ∈ Z" or "∃ x ∈ Z" tells us x is an integer—and it also states (b) whether we are talking about all values (i.e., every value) of x (∀) or (at least) one value of x (∃).
  2. The second part is the property of that variable. For instance, "x is even", or "x² ≤ 4".
  
  For clarity we normally enclose the quantification segment in parentheses (or square brackets [] or curly brackets {}), and also (usually) enclose the property segment in parentheses. For instance, "(∀ x ∈ Z) (x is even)". (That happens to be false, but true or false, the structure of the statement is the same).
• The property of x can be complicated. It may include a quantified statement about another variable. For instance,
                  "(∀ x ∈ Z) [ (∀ y ∈ N) (x > y) ]",
  or
                  "(∀ x ∈ Z) [ x < 1000 AND [(∃ y ∈ N) (x > y)] ]",
  but no matter how complicated, it is a property of x.
• One reason quantification segments must be in order is that a statement with a quantifier should have one variable with one quantification segment and one property segment. The property segment may contain its own quantified statement, whose property can contain a third quantified statement, etc. These multiple statements are always nested within one another, so there's a natural order from inside to outside (or outside to inside, if you prefer).
• As the book says, there's a simplification when two (or more) variables have the same quantifier (universal, or existential). Then the order of quantification doesn't affect the meaning, so the quantification segments can be combined. However, that's just to make it more convenient for us. It may still be helpful to rewrite the statement according to the basic structure, where only one variable is quantified at a time (especially when negating the statement).
• Sometimes, it's possible to omit the brackets without causing confusion. If you're not sure how to interpret a quantified statement, you should start by figuring out where the brackets ought to be.

• A is called the domain of f, written Dom(f), and B is its codomain, written Codom(f) (as in the book, except the notation).
• The book gives two definitions that are equivalent. In class we will discuss why they are equivalent.
• The set Im(f) := { f(x): x ∈ A } is called the image (or range) of f. This is the set of values actually taken on by f; it is an important set.
• We say f is injective (or one-to-one) if it never takes the same value twice. It is surjective (or onto) if Im(f) = B, i.e., f takes on every value in B. It is bijective (or a one-to-one correspondence) if it is both injective and bijective.

• Theorem 10.25A. Every real number r such that r > 1 has a square root, by which we mean a real number s such that s² = r.

• Theorem 10.25B. Every real number r such that 0 < r < 1 has a square root, by which we mean a real number s such that s² = r.

(Late correction.) The following is wrong: (N.B. These should be numbered 10.26A, 10.26B, 10.26C instead of 25, as they belong logically after Theorem 10.26.) The following is correct: They cannot be proved using 10.26. They are the proof of 10.26.