Foundations of Mathematics - Lexicographically - Mathematics Assignment Help

Assignment Task:

Foundations of Mathematics Assignment Help

Q.1 (from FoM, 6.B.13)
Prove any or all of the properties of unordered pairs (doubletons and singletons): (here a, b, c and d are assumed to be sets)
(i) {a, b} = {c, d} if and only if (a = c ∧ b = d) or (a = d ∧ b = c) .
(ii) {a} = {b} if and only if a = b .
(iii) {a} = {b, c} if and only if a = b = c .
(iv) Are any of these true (i.e. theorems) if we remove the assumption that a, b, c and d are sets?

Q.2 (from FoM, 6.E.2)
Sequences of natural numbers can be ordered “lexicographically from the left” by the definition: (writing x for
hx1, x2, . . .i and y for hy1, y2, . . .i )

x < y ⇔ there exists k such that xk < yk but for all i < k, xi = yi
.Similarly, we can order ”lexicographically from the right” by

x < y ⇔ there exists k such that xk < yk but for all i > k, xi = yi
.

In either case we define x ≤ y as usual to mean either x < y or x = y. These definitions work for both finite and infinite sequences. For each of the following kinds of sequences and order, decide which is the best you can say of the order: partial order, full order or well order (there is also the possibility that it may be none of these
things.)

(i) Nn , all sequences of some fixed length n, lexicographically ordered from the left.

(iiL) N ∞, all infinite sequences, lexicographically ordered from the left.

(iiR) N ∞, all infinite sequences, lexicographically ordered from the right.

Q.3
Consider these statements:
(a) A function f : A → B is surjective (onto) if and only if there exists a function g : B → A such that f ? g = idB .

(a) A function f : A → B is injective (one-to-one) if and only if there exists a function g : B → A such that g ? f = idB .

One of these statements is correct, the other is almost but not quite correct. Fix up the almost correct one and prove both. (If you use the Axiom of Choice, say so.)

Q.4
In FoM, Section 6.G.1 we prove that every vector space has a basis. Prove the stronger result that, if in a vector space, L is a linearly independent set, S is a set which spans the space and L ⊆ S, then there is a basis B for the space such that L ⊆ B ⊆ S. [A modification of the proof given in 6.G will do. Also you can use any result proved there.]

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