Exam (elaborations)
MAT2611 EXAM PACK 2023
- Course
- Institution
MAT2611 EXAM PACK 2023
[Show more]
Seller
Follow
pablitoh11
Reviews received
Content preview
MAT2611EXAM PACK
2023
QUESTIONS
AND
ANSWERS
, MAT2611/201/1/2020
LINEAR ALGEBRA
MAT2611
Department of Mathematical Sciences
This tutorial letter contains solutions for assignment 01.
BARCODE
university
Define tomorrow. of south africa
,Problem 1. Answer Exercise 2.22 from Addendum C. [10 marks]
Solution 1. The power set P(A) of a set A is the set of all subsets of A, i.e. we have
T ∈ P(A) ⇔ T ⊆ A.
Since ∅ ⊆ A (always, Why?)
and A ⊆ A always, the set P(A) always has at least the members ∅
and A, always.
As for example, since T ⊆ ∅ ⇔ T = ∅, we have:
P(∅) = {∅}.
Similarly, since T ⊆ {∅} if and only if either T = ∅ or else T = {∅}, we have:
P({∅}) =∅, {∅}
.
Similarly:
P({{∅}}) = ∅, {{∅}}
P({∅, {∅}}) = ∅, {∅}, {{∅}}, {∅, {∅}}
The rest can be figured out now.
Guess the number of elements of P(A) if A has exactly n elements and prove your guess.
Problem 2. Answer Exercise 3.12 & Exercise 3.13 from Addendum C. [5 + 5 = 10 marks]
f
Solution 2. Recall that X→
− Y if the following three conditions are satisfied:
(a) f ⊆ X × Y
(b) For each p ∈ X there exists a q ∈ Y such that (p, q) ∈ f .
0
(c) If (p, q) ∈ and
f (p, q) ∈ fthen q = 0q
.
f
Given X−
→ Y to be a one-to-one correspondence there is the additional property:
(d) For each q ∈ Y there exists a unique p ∈ X such that (p, q) ∈ f .
Hence the set:
(?) f −1 = (y , x ) : (x , y ) ∈ f
satisfies all the conditions (a)-(d) with X and Y interchanged.
−1
Verification for (a) From the definition in (?): (x , y ) ⇔
∈ (y
f , x) ∈ ⊆
f X × Y ⇒ (y, x) ∈ Y × X .
−1
Hence f ⊆ Y × X.
Verification for (b)&(c) Choose and fix any q ∈ Y . Using (d), for each q ∈ Y there exists a uniq
p ∈ X such that (p, q) ∈⇔ f(q, p) ∈ −1
f .
Hence for each q ∈ Y there exists a unique p ∈ X such that(q, p)∈ f −1 verifying the
conditions (b) & (c).
2
, MAT2611/201/1/2020
Verification for (d) For each p ∈ X there exists by (c) for
a unique
f q ∈ Y such that (p, q) ⇔
∈f
(q, p) ∈ −1 −1
f , verifying (d) for f.
The proof of f −1
◦f = 1Y and f−1◦f = 1X should now be clear from (a)-(d) for both
and ff−1.
[Total: 20 marks]
3
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller pablitoh11. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $2.90. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
57727 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now