Discrete Mathematics Answers

Use the Pigeonhole Principle to show that if there are 31 students in a class, then at least two have first names that begin with the same letter.

Calculate the number of one-to-one functions there are from a set with 6 elements to sets with the following numbers of elements:
(a) 5
(b) 6
(c) 7
(d) 8

Compute the number of functions f from the set {0,1,2,...; n} (where n is a positive integer), to the set {0,1}.

(2) How many different license plates can be made consisting of 10 characters if the first 5 characters consist of uppercase English letters, while the final 5 characters are digits?

Calculate the number of bit strings of length 5 or less.

In a class has 124 students, of these students 47 are involved in math club, 25 involved in science club and 36 involved in English club so how many are involved in both physics and English club?

Function f, g, h are defined on a set X = (1, 2, 3) as f = ((1, 2), (2, 3), (3, 1)}
g = {(1,2),(2,1),(3,1)} h = ((1,1), (2,2),(3,1)} i) Find fog, gof. Are they equal? ii) Find fogoh and fohog. Define partial function

Let R be a binary relation on the set of all positive integers such that R = { (a,b) | a- b is an odd positive integer } Is R reflexive, symmetric, antisymmetric, transitive? Is R an equivalence relation? A partial ordering relation

a) Let A = (1,2,3,4) and R = ((1 ,2),(2,4),( 1,3 ),(3 ,2)}. Find the transitive closure of R by Warshall’s algorithm.
b) Let A = {a,b,c}. show that (P(A),c ) is a poset and draw its Hasse diagram.
c) Define Chains and Antichains.

Draw the graph and its equivalent Hasse diagram for divisibility
on the set {1, 2, 3, 6, 12, 24, 36, 48}