Question

Read Proposition 1.23, the paragraph between the proposition and its proof, and the proof.
a. In your own words, explain the meaning of the two different mathematical ideas of "existence" and "uniqueness".
b. How do the ideas of "existence" and "uniqueness" relate to the phrase "one and only one"?
c. In your own words, explain why the proof of a "one and only one" statement will have two distinct parts.

Prove Proposition 1.26. Proposition 1.26. Let m,n ∈ Z. If mn = 0, then m = 0 or n = 0.

Prove Proposition 2.3. Proposition 2.3. 1 ∈ N.

Answer 1

a)

Existence: Existence means that there should exists at least one quantity satisfying the requirement either it to be a differential equation or inequality or equation etc,

Uniqueness:Uniqueness means that the quantity which exists should be unique i.e., there is only one quantity that satisfies the given condition or desired requirement.

b)

Existence means there should be a value exist and uniqueness means there should a quantity exist and that should be unique.Which we can simply say "one and only one",

c)From the (b) we see that the one and only one includes existence and uniqueness. So there is need to prove two parts one for existence and the other for the uniqueness while proving a theorem or statement.