Combinatorics and Logic Quiz

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What does the notation $\binom{n}{r}$ represent?

The binomial coefficient, which is the coefficient in the expansion of $(a + b)^n$. (correct)

The number of permutations of n objects taken r at a time.

The factorial of n divided by the factorial of r

The product of n and r

In Pascal's Triangle, how is each number (other than the 1's at the edges) generated?

By multiplying the two numbers appearing directly above it.

By adding the two numbers appearing directly above it. (correct)

By adding the two numbers appearing diagonally below it.

By subtracting the numbers above from its row number.

What is a permutation of a set of objects?

Any selection of objects from the set, regardless of order.

The total number of objects in the set.

A specific combination of r objects from n objects without repetition.

An arrangement of the objects taken all at a time or a subset taken at a time, in a given order. (correct)

How many 'factors' are present in both the numerator and the denominator of the binomial coefficient $\binom{n}{r}$?

r (C) Signup and view all the answers

Using the formula for permutations, how many different three-letter arrangements (i.e., 3-permutations) can be formed from a set of 6 objects?

120 (B) Signup and view all the answers

What does the binomial theorem state?

The expansion of a binomial raised to the power of n. (C) Signup and view all the answers

Which of the following is NOT a property of Pascal's Triangle?

Every other number can be obtained by dividing the two numbers appearing above it. (B) Signup and view all the answers

What is the formula for calculating the number of permutations of $n$ objects taken $r$ at a time, $P(n, r)$?

$P(n, r) = n(n-1)(n-2)...(n-r+1)$ (D) Signup and view all the answers

When is the biconditional statement $p \leftrightarrow q$ true?

When $p$ and $q$ have the same truth values. (A) Signup and view all the answers

Under what condition is the conditional statement $p \rightarrow q$ considered false?

When $p$ is true and $q$ is false. (A) Signup and view all the answers

According to the information provided, which of the following is logically equivalent to $p \rightarrow q$?

$\neg p \lor q$ (A) Signup and view all the answers

What defines a valid argument?

The conclusion is true whenever all the premises are true. (D) Signup and view all the answers

What is a fallacy in the context of arguments?

An argument where the conclusion is not true when all premises are true. (C) Signup and view all the answers

Given the argument $p, p \rightarrow q \vdash q$, what is the rule called?

Law of Detachment (C) Signup and view all the answers

Which argument represents a fallacy?

$p \rightarrow q, q \vdash p$ (B) Signup and view all the answers

When is the proposition $P_1 \land P_2 \land ... \land P_n$ true regarding premises?

When all of $P_1, P_2, ..., P_n$ are true simultaneously. (C) Signup and view all the answers

Which of the following statements accurately describes a symmetric relation R on a set A?

If (a, b) is in R, then (b, a) must also be in R. (B) Signup and view all the answers

A relation R on a set A is considered antisymmetric if:

Whenever (a, b) is in R and (b, a) is in R, then a equals b. (D) Signup and view all the answers

What condition must be met for a relation R on set A to be considered transitive?

If (a, b) is in R and (b, c) is in R, then (a, c) must be in R. (C) Signup and view all the answers

Given relation $R_1 = {(1, 1), (1, 2), (2, 2)}$, which of the following properties apply?

Antisymmetric and Transitive only (B) Signup and view all the answers

Given relation $R_2 = {(1, 1), (1, 2), (2, 1), (2, 2)}$, which of the following properties apply?

Symmetric and Transitive only (B) Signup and view all the answers

If a relation R is not symmetric, what must be true?

There exists at least one pair (a, b) in R such that (b, a) is not in R. (A) Signup and view all the answers

Consider the "less than or equal to" relation ($\leq$) on the set of integers. Which property does this relation satisfy?

Transitive, but not Symmetric (C) Signup and view all the answers

When considering the properties of a relation, what does closure refer to?

The smallest relation that contains a certain property. (C) Signup and view all the answers

What is a primary characteristic of a graph's adjacency matrix?

It is a symmetric matrix, with entries indicating the adjacency between vertices. (B) Signup and view all the answers

When is it generally more advantageous to use linked lists over an adjacency matrix to represent a graph?

When the graph is sparse, having significantly fewer edges relative to the number of vertices. (B) Signup and view all the answers

How does changing the order of vertices impact the adjacency matrix of a graph?

It results a different adjacency matrix, where rows and columns are interchanged. (A) Signup and view all the answers

What does the value 𝑎ᵢⱼ represent in an adjacency matrix of a multigraph?

The number of edges between vertices i and j. (B) Signup and view all the answers

In a weighted graph, what does 𝑎ᵢⱼ usually represent in the adjacency matrix?

The weight of the edge connecting vertices vᵢ and vⱼ. (C) Signup and view all the answers

What is a limitation of using an adjacency matrix to represent graphs?

It may be complex to insert or delete vertices. (A) Signup and view all the answers

What is a key feature of the 'linked representation' of graphs?

It uses linked lists of neighbors to represent adjacencies of vertices. (B) Signup and view all the answers

When is a graph considered 'dense'?

When the number of edges is quadratic, O(n²), with respect to the number of vertices. (A) Signup and view all the answers

In a Depth-First Search (DFS) algorithm, what data structure is primarily used for backtracking?

A Stack (C) Signup and view all the answers

What is the main purpose of the STATUS field used in both DFS and Breadth-First Search (BFS) algorithms?

To avoid processing vertices more than once (B) Signup and view all the answers

In a Breadth-First Search (BFS) algorithm, which data structure is used to keep track of neighbors waiting to be processed?

A Queue (A) Signup and view all the answers

If DFS processes vertices in the order A, D, L, K, C, J, M, B, what can be said about the order the neighbors of A were added to the stack?

B, then C, and finally D (B) Signup and view all the answers

How does the order of edge addition differ between the spanning trees generated by DFS and BFS?

BFS adds edges by level of reach, DFS moves directly to furthest neighbors first. (B) Signup and view all the answers

Given that BFS processes vertices in the order A, B, C, D, K, L, J, M, what is the order that vertex A's neighbors were added to the queue?

B, C, then D (C) Signup and view all the answers

What does a 'dead end' signify in the context of a Depth-First Search (DFS) algorithm?

A vertex that has no unprocessed neighbors. (D) Signup and view all the answers

How do the edges of the spanning trees created by both DFS and BFS relate to the original graph G?

Each edge corresponds to a connection found exploring an adjacency list. They only consist of a sub set of the edges found in G. (C) Signup and view all the answers

In a directed graph, what does the outdegree of a vertex represent?

The number of edges beginning at the vertex (B) Signup and view all the answers

If a vertex in a directed graph has an indegree of zero, it is considered a:

Source (C) Signup and view all the answers

Given a graph G with vertices V and edges E, a subgraph H is formed with vertices V' and edges E'. Which statement is always true?

The endpoints of edges in E' belong to V' (C) Signup and view all the answers

What condition defines a 'sink' vertex in a directed graph?

It has zero outdegree. (D) Signup and view all the answers

In a directed path, the direction of edges must:

Agree with the path’s direction. (A) Signup and view all the answers

If a directed graph has 10 edges, and the sum of all its vertex outdegrees is 10, what is the sum of all the indegrees of all the vertices?

10 (B) Signup and view all the answers

In the context of the ball-throwing example, if player B throws the ball to players A and C with equal probability, and player C also throws to A and B with equal probability, which statement about these probabilities is correct?

The sum of probabilities from B and C to A equals 1 (B) Signup and view all the answers

Consider a directed graph. If a subgraph H is 'generated' or 'determined' by a set of vertices V', what is a key feature of the edges in H?

They are all the edges whose end points belong to V'. (A) Signup and view all the answers

Study Notes

Discrete Structures (CSC203)

Learning Outcomes: Students should be able to convert logical statements, describe the strengths and limitations of propositional and predicate logic, outline proof techniques (direct, contradiction, and induction), apply these techniques, apply the pigeonhole principle, compute permutations and combinations, map real-world applications to counting, and solve recurrence relations.

Sets and Set Theory

A set is a collection of objects or numbers
Membership in a set is denoted by ∈
Sets are specified by listing elements or by listing properties
A subset is a set whose elements are all elements of another set (A⊆B)
Two sets are equal if they contain the same elements (A=B if and only if A⊆B and B⊆A)
Disjoint sets have no elements in common (A∩B=Ø)
Universal set (U) is the largest set containing all sets under consideration
Empty set (Ø or {}) is the set with no elements
Venn diagrams are pictorial representations of sets

Set Operations

Union (∪): The set of all elements in either set A or set B (or both). A ∪ B = {x | x ∈ A or x ∈ B}
Intersection (∩): The set of elements common to both sets A and B. A ∩ B = {x | x ∈ A and x ∈ B}
Complement (^c): The set of elements in the universal set that are not in set A. A^c = {x ∈ U | x ∉ A}
Difference (∖): The set of elements in A but not in B. A ∖ B = {x | x ∈ A and x ∉ B}
Symmetric Difference (⊕): The set of elements in either A or B, but not both. A ⊕ B = (A ∖ B) ∪ (B ∖ A)

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Test your understanding of combinatorial concepts, including binomial coefficients, permutations, and the properties of Pascal's Triangle. This quiz also covers basic logical statements and their equivalences. Perfect for students studying probability and logic in mathematics.