CS 131 UNH Using Intersection Union Definition & Contradiction Worksheet

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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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Tags:
universal quantifier

logical statement

contradiction

union definition

using intersection

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