Discrete Mathematics Answers

Define a relation - on the set of real numbers by

x-ymeans I x | + I y I : I x+y l.

which of the properties for an equivalence relation does - satisfy?

Draw the Hasse diagram for the lattice (D₂4, 1). Hence find. a) maximal & minimal element b) glb of 446 geb of 8 & 2 d) lub of 8 & 6

Consider a relation R=\ (1,1),(1, ), (0,2), (2,3) (3,1)) on the set A=\ 1,2,3\ Find transitive closure of the relation R using algorithm Warshall's

For the relation R = {(p,p) ,(q,p),(q,q),(r,r),(r,s),(s,s) ,(s,m) ,(m,m)}

1.Using warshall algorithm find the transitive closure R* of R

2.write matrix representation of R*

3.Check whether the relation of R* is an equivalence relation or a partial order.

A pet store keeps track of the purchases of customers over a fours hour period. The store manager classifies purchases as containing a dog product, a cat product, a fish product, or product for a different kind of pet. He found!

83 purchased a dog product

101 purchased a cat product

22 purchased a fish product

31 purchased a dog and a cat product

8 purchased a dog and a fish product

10 purchased a cat and a fish product

6 purchased a dog, a cat, and a fish product

34 purchased a product for a pet other than a dog, cat, or fish.

Draw a Venn diagram to find that:

(1) How many purchases were for a dog product only?

(ii) How many purchases were for a cat product only?

How many purchases were for a dog or a fish product?

(iv) How many purchases were there in total?

Let a and b be two cardinal numbers. Modify Cantor’s definition of a < b to define a ≤ b. (Hint: Examine what happens if you drop condition (a) from Cantor’s definition of a < b.) 2. Prove that a ≤ a. 3. Prove that if a ≤ b and b ≤ c, then a ≤ c. 4. Do you think that a ≤ b and b ≤ a imply

a = b? Explain your reasoning. (Hint: This is not as trivial as it might look.)

Show that a lattice is distributive if and only if for any elements a,b,c in thee lattice (aVb) V c< aV (bVc)

show that ~Q,P—>Q=>~P in mathematical foundations of computer science

The English alphabet contains 21 consonants and fivevowels. How many strings of six lowercase letters of theEnglish alphabet contain:

a) exactly two vowels?

b) at least two vowels?

Find the sum-of-products expansions the Boolean function F(x, Y, 2) that equals 1 if and only if

a)x = 0.

b) xy = 0.

c) x +y = 0.

d)xyz = 0.