MAT246: Reading Assignment 2
Due: Friday, May 23rd at 11:59PM
Instructions: Complete this assignment independently after watching/reading any pre-class material, either by writing
directly on the PDF with a tablet, or by printing it and writing directly on your printed copy. Afterwards, submit to
Gradescope. If you print it, please scan using Evernote Scannable for iOS, or Genius Scan for Android, and ensure
that the boxes line up in Gradescope.
Please remember to include sucient reasoning with your answers in order to earn full marks.
Problem 1. Carefully prove using the definitions that if A is a set, then A ✓ A and ? ✓ A.
Problem 2. If A and B are subsets of some universal set U , prove carefully that (A [B)c = Ac \Bc.
wrz ; A set⇒ AEA ψE ⼋
UX . LXEA =I xEA ) By dekmition ot subset .
=7 A EA
VxILOEA 3xLA )iydefpmittin ot sub sets .
=⇒ ψ EA
We hare15 is aset⇒ AEA M LEA
⽹第
“ Snce AUB 些 xEAVxcBlix
Asamddet {hLxE 仔 }
⇒ AUBCJ et TxI xEA xEB }bg logial equivalonce
of ngation .
Eince AC ht IXI xEAl iBlotIelaaBJd
=3 AC n 13
' ht |xlxtA xtBIM
CAUB
)
=ANB
Problem 3. Suppose that A, B, and C are sets. Prove that if A ✓ B and B ✓ C, then A ✓ C.
Problem 4. For each n 2 N, define An = {x 2 R | n < x < n+ 1}. Write the set
[
n2N
An as the di↵erence of two sets.
Prove that they are equal from the definitions.
Problem 5. In a certain village, there is a male barber who shaves every man who does not shave himself. Explain
why this is a paradox and how it is related to Russell’s paradox.
Shee A , 13 , C are sets coul AEB oamd BEC
By definiliou , Ux , CxE AH⇒ xE 13 ) anl Vx.
(xEB =) x ε l )
then
,
VC
.
IxEA )xE( 3 =3 xtC )
=⇒ θa . < xEA =) xEC )
=⇒ AEC
Ytrtto ef Tl 雪 BEIN . xEAB}
In thhiscase, ofthemaleburbershivehimselt ,thenheshould not shave himselt
,
Becausebythedethitlen
,theberberconntshhrethemolewhoshae himselt which is
the burber himse lt . C a con tradiction)
Homever , itthemaleborbesdoes not shavehinselt, thenheshouldshove himselt
.
Beccuse
by
thedefinition
,
theborbershold shoive every mon who doe, n -t shave hninself which is
the barber himself . Ca coatra diction
=⇒ This is a Russe lls paradox .