WORKSHEET 1: PRACTICE ON SET THEORY
MATH 2106-D
(1) Let X={1,5,8}and Y={1, a, b}. Write down the following sets:
(a) X∩Y.
(b) X∪Y.
(c) P(X∩Y).
(d) X×Y
(2) If Iis a set, called an index set, and for each α∈ I, we have a set Aα, then we
can consider the union and intersection
∪α∈I Aα={x|x∈Aαfor at least one Aαwith α∈ I} ,
∩α∈I Aα={x|x∈Aαfor every Aαwith α∈ I} .
Determine what the following sets are, and explain your answer.
(a) ∩n∈N−1
n,1
n
(b) ∪n∈N−1
n,1
n
(c) ∪a∈R{(x, y, z)|x2+y2=a2, z =a}
(3) Explain why |A∪B|=|A|+|B| − |A∩B|. Can you find a similar formula for
|A∪B∪C|in terms of cardinalities of A, B, C and intersections between these
sets?
(***) Let Xbe the set of all sets that are not elements of themselves, that is,
X={A|Ais a set, A 6∈ A}.
Explain why the existence of this “set” doesn’t make logical sense. (Hint: Is
X∈X?) This conundrum, known as Russell’s Paradox, shows that in order
to be 100% rigorous with set theory, one must be very careful when describing
which sets are “allowed”; the standard resolution to this problem is to base set
theory on a precise set of axioms, such as ZFC.
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