Discrete Structures โ Fall-2020 Assignment-01 Due Date: 18 Nov 2020
Counting
Problem 1
From section 1.1 of the given reading, what is the number of pairings (in all the various senses as discussed) in a
party of 12?
a. Handshakes b. Seating arrangements c. Playing chess (with 6 chess-boards)
Problem 2
(a) How many words of 8 letters can you form, using the English alphabet? (there are 26 different letters, of which 5
are vowels).
(b) How many words of length 8 can you form, where no letter is repeated?
(c) How many words of length 8 can you form, where the ๏ฌrst letter is the same as the last letter?
(d) How many words of length 8 can you form, which contain exactly one vowel?
Problem 3
In how many ways can a photographer at a wedding arrange 8 people in a row, including the bride and groom, if
a) the bride must be next to the groom?
b) the bride is not next to the groom?
c) the bride is positioned somewhere to the left of the groom?
Problem 4
In how many ways can you seat 12 people at two round tables with 6 places each?
Think of possible ways of defining when two seatings are different, and find the answer for each.
Sets
Problem 5
a. List all subsets of {๐ฅ, ๐ฆ,๐ง}. How many do you get?
b. List all subsets of {๐ฃ,๐ค,๐ฅ,๐ฆ,๐ง}, containing ๐ฆ but NOT containing ๐ง.
c. Define a set of which both {1,3, 4} and {0,3,5} are subsets. Find such a set with the smallest possible number
of elements.
d. What is the number of subsets of a set with n elements, containing a given element (one element becomes
fixed; it is part of every subset)?
Problem 6
a. We form the union of a set with 5 elements and a set with 9 elements. Which of the following numbers can
we get as the cardinality of the union: 4, 6, 9, 10, 14, 20?
b. We form the union of two sets. We know that one of them has n elements and the other has m elements. What
can we infer about the cardinality of their union?
c. What is the intersection of
i) the sets { } and {1,2, 3};
ii) the set of girls in this class and the set of boys in this class;
iii) the set of prime numbers and the set of even numbers?
d. We form the intersection of two sets. We know that one of them has n elements and the other has m elements.
What can we infer about the cardinality of their intersection?
Problem 7
Use Venn-diagram to prove following:
a. Prove equations (1.2), (1.3), and (1.4) given in the reading pages.
b. Prove that |๐ด โช ๐ต| + |๐ด โฉ ๐ต| = |๐ด| + |๐ต|.
Problem 8
a. What is the symmetric difference of the set โค+ of nonnegative integers and the set ๐ผ of odd integers (๐ด =
{. .. , โ3, โ1,1, 3, .. .} contains both negative and positive odd integers).
b. Let ๐ถ be the symmetric difference of ๐ด and ๐ต ( that is ๐ดฮ๐ต = ๐ถ). Now, form the symmetric difference of A
and C. What did you get? Give a proof of the answer using Venn diagram.
Problem 9
Prove the extended version of binomial identity. Prove by both methods:
i. Algebraic way (evaluate the given expression)
ii. By combinatorial argument of counting
(๐
๐)= (๐ โ 2
๐ โ 2)+2(๐ โ 2
๐ โ 1)+(๐ โ 2
๐)
