Question

Let S be a collection of subsets of [n] such that any two subsets in S have a non-empty intersection. Show that |S| ≤ 2^(n−1).

Answer #1

Proof attached

Let S be the sample space of an experiment and let ℱ be a
non-empty collection of subsets of S such that i) ? ∈ ℱ ⇒ ? ′ ∈ ℱ
and ii) ?1 ∈ ℱ and ?2 ∈ ℱ ⇒ ?1 ∪ ?2 ∈ ℱ
a) Show that if ?1 ∈ ℱ and ?2 ∈ ℱ then ?1 ∩ ?2 ∈ ℱ .
b) Show that ? ∈ ℱ.
c) Is ℱ necessarily a ?-algebra? Explain briefly. A rigorous...

Let f : X → Y and suppose that {Ai}i∈I is an indexed collection
of subsets of X. Show that f[∩i∈IAi ] ⊆ ∩i∈I f[Ai ]. Give an
example, using two sets A1 and A2, to show that it’s possible for
the LHS to be empty while the RHS is non-empty.

Let X be a set and let (An)n∈N be a sequence of subsets of X.
Show that: (a) If (An)n∈N is increasing, then liminf An = limsupAn
=S∞ n=1 An. (b) If (An)n∈N is decreasing, then liminf An = limsupAn
=T∞ n=1 An.

Let {??}?∈? be an indexed collection of subsets of a set ?.
Prove:
a. ?\(⋃ ??) ?∈? = ⋂ (?\??) ?∈?
b. ?\(⋂ ??) = ⋃ (?\??) ?∈?? ∈?
Note: These are DeMorgan’s Laws for indexed collections of
sets.

Let n ≥ 2 be any natural number and consider n lines in the xy
plane. A point in the xy plane is called an intersection point if
at least two lines pass through it. Use induction to
show that the number of intersection points is at most
(n(n-1))/2.

Let A, B be non-empty subsets of R. Define A + B = {a + b | a ∈
A and b ∈ B}. (a) If A = (−1, 2] and B = [1, 4], what is A + B?

Suppose that the set S has n elements and discuss the
number of subsets of various sizes.
(a) How many subsets of size 0 does S have?
(b) How many subsets of size 1 does S have?
(c) How many subsets of size 2 does S have?
(d) How many subsets of size n does S have?
(e) Clearly the total number of subsets of S must equal the sum of
the number of subsets of size 0, of size...

Let S be a finite set and let P(S) denote the set of all subsets
of S. Define a relation on P(S) by declaring that two subsets A and
B are related if A and B have the same number of elements.
(a) Prove that this is an equivalence relation.
b) Determine the equivalence classes.
c) Determine the number of elements in each equivalence
class.

1)Let the Universal Set, S, have 97 elements. A and B are
subsets of S. Set A contains 45 elements and Set B contains 18
elements. If Sets A and B have 1 elements in common, how many
elements are in A but not in B?
2)Let the Universal Set, S, have 178 elements. A and B are
subsets of S. Set A contains 72 elements and Set B contains 95
elements. If Sets A and B have 39 elements...

Let S and T be nonempty subsets of R with the following
property: s ≤ t for all s ∈ S and t ∈ T.
(a) Show that S is bounded above and T is bounded below.
(b) Prove supS ≤ inf T .
(c) Given an example of such sets S and T where S ∩ T is
nonempty.
(d) Give an example of sets S and T where supS = infT and S ∩T
is the empty set....

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