Question

Find the power set of the following.
[i] A=∅
[ii] B={1,2}
[iii] C={p,q,r}
NO OPTIONS PLEASE

Answer 1

Answer:

1. Nothing

2. 1 and 2

3. p,q and r

Answer 2

Answer:

What is the cardinality of each of these sets?

a) ∅ 0

b) {∅} 1

c) {∅, {∅}} 2

d) {∅, {∅}, {∅, {∅}}} 3

Problem Three (1.6.16)

Can you conclude that A = B if A and B a e two sets with the same power set?

Yes. By definition, P(A) is the set of all subsets that can be generated from A, if A and B generate the exact same collection

of valid subsets, then it must be that A and B contain the same elements and are therefore equal.

Problem Four (1.6.20)

What is the Car esian product A cross B, where A is the set of cou es offered by the mathematics department at a

university and B is the set of mathematics p ofessors at this unive ity

It is the set of all possible combinations of math courses and possible instructors.

Problem Five (1.7.4)

Let A={a, b, c, d, e} and B={a, b, c, d, e, , g, h}. Find

a) A ∪ B {a, b, c, d, e, f, g, h}

b) A ∩ B {a, b, c, d, e}

c) A – B { }

d) B – A {f, g, h}

Problem Six (1.7.16)

Show that, i A and B are sets, then (A ∩ B) ∪ (A ∩ BC) = A. [Note: BC is another way of writing the complement of set B]

There are two ways of solving set proofs like these, one is to look at an arbitrary point and use the properties of sets to

argue why something it true. The other way to do this is to notice that a set is nothing but a collection of elements, and

that collection of elements will be the truth set for some proposition. Therefore, if we can prove that the truth sets on

either side of the equation are equal, then the sets they represent must also be equal. Both proof forms are provided as a

solution. Note that, while the proposition-style proof is inherently longer, it is often far more formulaic in its construction

and therefore easier to write and, usually, understand.

Set-style Proof

Note that every point a, in A, is an element of either A ∩ B (in the case that a∈B) or an element of A ∩ BC (in the case that

a∈BC). Also, if a∈(A ∩ B) ∪ (A ∩ BC), then either a∈A ∩ B or a∈ ∩ BC, in either of these cases, a∈A by the definition of

intersection, so it must be that (A ∩ B) ∪ (A ∩ BC) = A.

Proposition-style Proof

Let p(x) be the proposition whose truth set is the set A

Let q(x) be the proposition whose truth set is B