Discrete Mathematics Answers

X be a non-empty set and let R be an equivalence relation on X. For each x ∈ X, define
[x]={y∈X suchthatxRy}
to be the equivalence class of x. Here x R y means (x, y) ∈ R.
SupposethatA=[x]andB=[y]. ProvethatifA∩B̸=∅,thenA=B.

Question one

Prove by induction that the following statements are true for all integers n
a) 12x2+22x3+…..+n2(n+1)= n(n+1)(n+2)(3n+1)/12
b)4007n-1 is divisible by 2003

Question five

For each of the following sentences, write the sentence in logical notation , negate the sentence , and say whether the sentence or its negation is true.
a) Given any integer, there is a larger integer.

Prove that A ∩ B = A ∪ B.

Find the truth set of each of these predicates where the
domain is the set of integers.
a) P(x): x3 ≥ 1 b) Q(x): x2 = 2
c) R(x): x < x2

Find the number of circular 3-permutations of 5 people.

Let f be the function from {a, b, c, d} to {1, 2, 3} defined by f(a) = 3, f(b) = 2, f(c) = 1, and f(d) = 3. Is f an onto function?

f is the function from {a, b, c} to {1, 2, 3} such that f(a)=2, f(b)=3, f(c)=1. Is f invertible, and if it is, what is its inverse?

A committee of three is chosen from a group of 20 people. How many different committees are possible, if

(a) the committee consists of a president, vice president, and treasurer?

(b) there is no distinction among the three members of the committee?

Let X be a finite set with |X| > 1. What is the difference between P1 = X ×X and P2 = {S ∈ P(X) | |S| = 2}? Which set, P1 or P2, has more elements?