Discrete Mathematics Exam II

A math exam covering discrete mathematics concepts.

An exam where students are not allowed to use books, notes, or other materials during the exam.

Questions with a set of possible answers, only one of which is correct.

The exam is scored out of 100 points, potentially with additional extra credit.

Distribute time effectively. Don't linger on a particular problem.

Examine questions for any new combinations or variations of previous homework problems.

The second exam in a discrete mathematics course.

Thursday, 25 April, 2024

The sum of two series, S1 and S2, where each term in S1 is a product of powers of 3 and 2. The sum of the series is given by ∑ (3+2)^j for j from 0 to n

The sum of two series, S1 and S2, where the terms in each series are powers of 3 and 2. For any j, S1 term = 3^j and S2 term= 2^j, The sum of the series is ∑ (3j + 2j) for j from 0 to n.

The sum of two series. S1 and S2, where the terms in S1 are the powers of 3 & S2 are powers of 2. In this case, The combined sum is ∑(3 * 2)^j, for j from 0 to n.

A sequence of numbers such that the difference between consecutive terms is constant.

A sequence of numbers such that the ratio between consecutive terms is constant.

A set that can be put into a one-to-one correspondence with the natural numbers.

A function that maps each element of set A to a unique element in the set of natural numbers N, while some elements of N may not be used.

A function that maps every element of the set of natural numbers N to some element in set A. Every element of A must be mapped to.

A function that is both injective and surjective, creating a one-to-one correspondence between two sets.

The cardinality of the set of integers (Z) is equal to the cardinality of the set of natural numbers (N).

The sum of the first n natural numbers (1 + 2 + 3 + ... + n) is equal to n(n + 1)/2

The sum of the first n+1 terms of a geometric series (a + ar + ar^2 + ... + ar^n) is equal to (ar^(n+1) - a)/(r - 1), given r ≠ 1.

The sum of 1 + 2 + 2^2 + ... + 2^n is equal to 2^(n+1) - 1

Positive odd integers (Zodd = {1, 3, 5,...}) are countably infinite, meaning they can be put in a one-to-one correspondence with the natural numbers.

The number of ways to seat n people around a circular table, where two arrangements are the same if each person has the same neighbors is (n-1)!.

The number of bit strings of length 10 that either begin with six 0s or end with five 0s or both is 2^4+2^5-2^3=16+32-8=40

For all positive integers k, if P(k) is true, then P(k+1) is also true.

The number of ways to arrange r items from a set of n items, where order matters and repetition is not allowed.

The number of ways to choose r items from a set of n items, where order doesn't matter and repetition is not allowed.

If a task can be done in n1 ways or n2 ways (mutually exclusive), then there are n1+n2 total ways to do the task.

If a task can be done in n1 ways and another task can be done in n2 ways, then both tasks can be done together in n1*n2 ways.

A function where each element in the domain maps to a unique element in the range. No two inputs map to the same output.

Choosing a password of length k from distinct items (no repetition), there are n * (n-1) * ... * (n - k + 1) possible ways. (n choose k, is a combination formula).

Find the number of bit strings of a certain length with no consecutive 9 1's.

The number of ways to choose an unordered team of 40 students from a class of 89 registered students