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web.groovymark@gmail.com
- November 15, 2024
Question 01
What does the term “bijection” mean in the context of functions?
a) A function that is both injective and surjective
b) A function that is neither injective nor surjective
c) A function that is injective but not surjective
d) A function that is surjective but not injective
Correct Answer: a) A function that is both injective and surjective
Explanation: A bijection is a function that is both injective (one-to-one) and surjective (onto), meaning every element of the domain maps to a unique element of the codomain, and all elements of the codomain are covered.
Question 02
Which of the following is true for a function to be injective?
a) Distinct elements in the domain map to distinct elements in the codomain
b) Multiple elements in the domain can map to the same element in the codomain
c) Every element in the codomain must be mapped by at least one element in the domain
d) The domain and codomain must have the same number of elements
Correct Answer: a) Distinct elements in the domain map to distinct elements in the codomain
Explanation: A function is injective if every distinct element in the domain maps to a distinct element in the codomain, meaning no two elements in the domain map to the same element in the codomain.
Question 03
What is the composition of two functions, f and g?
a) The application of g followed by f
b) The addition of the values of f and g
c) The subtraction of the values of f and g
d) The application of f followed by g
Correct Answer: a) The application of g followed by f
Explanation: The composition of two functions, denoted f(g(x)), involves applying function g first and then applying function f to the result of g.
Question 04
Which of the following is a property of a surjective function?
a) Every element in the codomain is mapped by at least one element in the domain
b) Every element in the domain maps to a unique element in the codomain
c) Some elements in the codomain are not mapped by any element in the domain
d) Every element in the domain maps to the same element in the codomain
Correct Answer: a) Every element in the codomain is mapped by at least one element in the domain
Explanation: A function is surjective (onto) if every element in the codomain is the image of at least one element from the domain.
Question 05
Which of the following statements about a function’s inverse is true?
a) The inverse undoes the effect of the original function
b) The inverse doubles the effect of the original function
c) The inverse negates the output of the original function
d) The inverse always returns the input multiplied by 2
Correct Answer: a) The inverse undoes the effect of the original function
Explanation: A function's inverse undoes the effect of the original function, meaning that applying the function and then applying its inverse will return the original input.
Question 06
Which of the following is true for a function to have an inverse?
a) The function must be both injective and surjective
b) The function must be neither injective nor surjective
c) The function must be injective but not surjective
d) The function must be surjective but not injective
Correct Answer: a) The function must be both injective and surjective
Explanation: For a function to have an inverse, it must be bijective, meaning it is both injective (one-to-one) and surjective (onto).
Question 07
Which of the following describes a predicate?
a) A logical statement whose truth value depends on one or more variables
b) A logical statement that is always true
c) A logical statement that is always false
d) A logical statement that contains no variables
Correct Answer: a) A logical statement whose truth value depends on one or more variables
Explanation: A predicate is a logical statement that involves variables, and its truth value depends on the values assigned to those variables.
Question 08
Which of the following is true for a relation to be an equivalence relation?
a) It must be reflexive, symmetric, and transitive
b) It must be reflexive and antisymmetric
c) It must be transitive and symmetric
d) It must be antisymmetric and irreflexive
Correct Answer: a) It must be reflexive, symmetric, and transitive
Explanation: An equivalence relation is a relation that is reflexive, symmetric, and transitive, meaning it satisfies these three properties.
Question 09
What is the difference between a reflexive and irreflexive relation?
a) A reflexive relation includes self-loops for every element, while an irreflexive relation has no self-loops
b) A reflexive relation has no self-loops, while an irreflexive relation includes self-loops
c) A reflexive relation is symmetric, while an irreflexive relation is antisymmetric
d) A reflexive relation is transitive, while an irreflexive relation is not
Correct Answer: a) A reflexive relation includes self-loops for every element, while an irreflexive relation has no self-loops
Explanation: A reflexive relation includes self-loops for every element in the set, while an irreflexive relation has no self-loops for any element.
Question 10
Which of the following is a key characteristic of a symmetric relation?
a) If a is related to b, then b is related to a
b) If a is related to b, then a is not related to b
c) If a is related to b, then a must equal b
d) If a is related to b, then a is not related to c
Correct Answer: a) If a is related to b, then b is related to a
Explanation: A relation is symmetric if whenever one element is related to another, the reverse relation also holds.
Question 11
Which of the following describes an antisymmetric relation?
a) If a is related to b and b is related to a, then a=b
b) If a is related to b, then a≠b
c) If a is related to b, then b is not related to a
d) If a is related to b, then a must equal b
Correct Answer: a) If a is related to b and b is related to a, then a=b
Explanation: A relation is antisymmetric if for all elements a and b, if a is related to b and b is related to a, then a=b.
Question 12
Which of the following describes the property of transitivity in a relation?
a) If a is related to b, and b is related to c, then a is related to c
b) If a is related to b, then a is also related to c
c) If a is related to b, then a must equal b
d) If a is related to b, then b must equal c
Correct Answer: a) If a is related to b, and b is related to c, then a is related to c
Explanation: A relation is transitive if, whenever a is related to b and b is related to c, it follows that a is related to c.
Question 13
Which of the following describes a directed graph?
a) A graph where edges have a direction from one vertex to another
b) A graph where all edges are undirected
c) A graph where each vertex is connected to all other vertices
d) A graph where edges form a cycle
Correct Answer: a) A graph where edges have a direction from one vertex to another
Explanation: A directed graph, also known as a digraph, has edges with directions, meaning that an edge from vertex a to vertex b is not necessarily the same as an edge from b to a.
Question 14
Which of the following is a property of an undirected graph?
a) All edges have no direction, meaning connections between vertices are mutual
b) All edges have a direction
c) All vertices are connected to each other
d) The graph contains a cycle
Correct Answer: a) All edges have no direction, meaning connections between vertices are mutual
Explanation: In an undirected graph, the edges do not have a direction, so if there is an edge between two vertices, the connection is mutual.
Question 15
Which of the following is true about a cycle in graph theory?
a) It is a path that starts and ends at the same vertex with no repeated edges
b) It is a path that starts and ends at different vertices with no repeated edges
c) It is a path that starts and ends at the same vertex with repeated edges
d) It is a path that has no edges
Correct Answer: a) It is a path that starts and ends at the same vertex with no repeated edges
Explanation: In graph theory, a cycle is a path that starts and ends at the same vertex without repeating any edges.
Question 16
Which of the following best describes a walk in graph theory?
a) A sequence of vertices connected by edges, where vertices and edges can be repeated
b) A sequence of vertices connected by edges, with no repeated edges
c) A sequence of vertices connected by edges, with no repeated vertices
d) A sequence of vertices where all vertices are adjacent
Correct Answer: a) A sequence of vertices connected by edges, where vertices and edges can be repeated
Explanation: A walk in graph theory is a sequence of vertices and edges where both vertices and edges can be repeated.
Question 17
Which of the following is true for a spanning tree?
a) It is a subgraph that contains all vertices of the original graph and is acyclic
b) It is a subgraph that contains some vertices of the original graph and is acyclic
c) It is a subgraph that contains all vertices and edges of the original graph
d) It is a subgraph that contains all cycles of the original graph
Correct Answer: a) It is a subgraph that contains all vertices of the original graph and is acyclic
Explanation: A spanning tree is a subgraph that contains all the vertices of the original graph, but it has no cycles.
Question 18
What is a leaf in a tree graph?
a) A vertex with no children
b) A vertex connected to all other vertices
c) A vertex that is part of a cycle
d) A vertex that has the highest degree
Correct Answer: a) A vertex with no children
Explanation: In a tree graph, a leaf is a vertex that has no children, meaning it has no further vertices connected to it.
Question 19
What is the height of a tree graph?
a) The number of edges from the root to the deepest leaf
b) The number of vertices in the tree
c) The number of vertices from the root to the first leaf
d) The number of children each vertex has
Correct Answer: a) The number of edges from the root to the deepest leaf
Explanation: The height of a tree graph is the length of the longest path from the root to any leaf, measured in the number of edges.
Question 20
What does it mean for a graph to be k-vertex connected?
a) The graph remains connected even after any k−1 vertices are removed
b) The graph has exactly k vertices
c) The graph has exactly k edges
d) The graph contains k cycles
Correct Answer: a) The graph remains connected even after any k−1 vertices are removed
Explanation: A graph is k-vertex connected if it remains connected after any k−1 vertices are removed, meaning it is resilient to the removal of vertices.
