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I really need a tutor who are good at solving set problem. I have finished my assignment, but I need this tutor to help me double check if the answer is correct. Attached is the questions which need your help, please showing your bids kindly if you think you could help me.

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Problem 2. [40 Points] Given a set S, its power set P(S) is defined to be the set of subsets of

S. For example, if S = {a,b}, then P(S) = {,{a},{b}, {a,b}}. Set operations can be expressed

as propositional logic formulas; for a set S and an element x, there is a corresponding Boolean

variable (x ES). Consider:

1. P(A) U P(B) C P(AUB)

2. P(A) n P(B) C P(An B)

3. P(A) U P(B) C P(AUB)

4. P(A) n P(B) C P(ANB)

and find out which one is a correct statement and which one is incorrect. Note that C means subset

and c means proper subset.

In order to show that an equation is False/incorrect (not always True), give an example of sets

A and B for which the equation does not hold. Provide a clear explanation why it doesn’t. In

order to show that an equation is always True/correct, prove that using definitions, the laws of

propositional logic, rules of inference as outlined in zy Books and lecture notes. You may also use

the following rule of inference, which is known the Conditional Conjunction.

A + B

A +C

.:. A – В ЛС

the laws of propositional logic and for practice consider

Note that this rule can be proved

proving it for yourself.

You may also want to use the fact that x e P(A) H x

H3 CA.

Problem 3. (10 points] We want you to express the following as a logical statement with quantifiers:

A is not an empty set

Please do so in two different ways: one way should use only the universal quantifier, and the other

should use only the existential quantifier.

Problem 4. [10 Points) Let’s suppose we are interested in writing some formal propositions about

a bunch of people, some of whom are students in a school (U) and a bunch of courses, some of

which are offered by U.

.

C(s) : s is an offered course at the U

• S(x) : x is a student at U

.

E(2,s) : student x is enrolled in a course s

Translate the following assertions using quantifiers:

a) Every student at U is enrolled in at least one course at U

b) There is a course at U that all students are enrolled in

Problem 5. (20 points] Given a finite set A, we denote its size (number of elements) as |A|. The

Jaccard similarity J(A,B) of the finite sets A and B is defined to be J(A, B) = |An B/|AU BI,

with J(0,0) = 1. The Jaccard distance dj(A, B) between A and B equals dj(A, B) = 1- J(A,B).

a) Find J(A, B) and dj(A.B) for these pairs of sets:

1. A = {1,3,5}, B = {1, 2, 4, 6}

2. A = 0, B = {0}

b) Prove that J(A, B) = 1 if and only if A = B.

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universal quantifier

logical statement

contradiction

union definition

using intersection

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