Understanding Sets, Relations, and Functions for Struggling Students

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Plonker1
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I'm having issues with the first four questions and have uploaded them. My attempts are shown below.

1.
a) True, all elements of E are even
b) False, 0 is not a multiple of 3
c) True, 8 is even and 9 is a multiple of 3
d) No idea
e) False, 6 is an element of E and T
f) No idea

2.
a) You can see my drawing attempt on imgur, sorry for the messiness (ed's note: see below). The parts meant to be coloured in are in black.
b) This claim is false
c) I drew up another graph that I guessed proves it is false. Am I correct?

3&4 I have no idea how to do.

Thank you so much for any help,
A struggling student.
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Attachments
  • Printed exam question 1+ defining sets U, E and T, with six true-or-false parts (a) to (f) on set membership, intersection, union, disjointness and power set.
    v1.png
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  • Question 2 from a set theory exam: (a) Draw a Venn diagram of four sets A, B, C, D where A ⊆ C and B ⊆ D; (b) decide truth of [(A ⊆ C) ∧ (B ⊆ D)] ⇒ [(A△B) ⊆ (C△D)]; (c) prove answer to (b).
    v2.png
    9.1 KB · Views: 175
  • Table of match results for six teams A to F, with Home teams as rows and Away teams as columns, showing W, L or blank for each pairing; beside it, a directed graph on vertices A to F with arrows between some of them.
    v3.png
    23.7 KB · Views: 167
  • Printed exercise Question 4 defining functions a, b, c, d from N x N to N: a(x,y)=xy, b(x,y)=x+y, c(x,y)=(x+y)^2+x, d(x,y)=(m-1)^2+m+x-y with m=max(x,y), asking whether each is injective, surjective or bijective.
    v4.png
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  • Handwritten notes with answers 1.c to 1.f and 2.a A∧B⇒C∧D, above a hand-drawn Venn-style diagram of overlapping shaded regions with two areas highlighted in yellow, and below it a set of overlapping loops.
    v5.jpg
    66.2 KB · Views: 176
  • Handwritten Venn-style diagram of three overlapping ellipses labelled with the expression 2. (b, c) A ⊕ B ⇒ C ⊕ D, with regions shaded in black hatching and bright green highlighter.
    v6.jpg
    52.3 KB · Views: 154
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I will comment on a couple things, but are you including 0 in your definition of the natural numbers?
Meaning, do you define the natural numbers as
N = {0,1,2,...} or N = {1,2,...}

Some people include 0 and some do not.