What does the union of two sets A and B represent?
A Set difference
B All elements
C Common elements
D Subset of A
The union of two sets A and B, denoted as A ∪ B, represents a set that contains all the elements from both sets, without repetition. It includes elements that are in A, B, or both.
In propositional logic, which of the following is a tautology?
A (A AND B) OR (NOT A)
B A AND NOT A
C A AND (A OR B)
D A OR (A AND B)
The expression A OR (A AND B) is a tautology because, for any values of A and B, the result is always true. If A is true, the entire expression is true, regardless of B’s value.
Which of the following is true for a relation R on set A?
A R contains elements from A only
B R is a subset of B
C R is a subset of A×A
D R is a subset of A
A relation R on a set A is defined as a subset of the Cartesian product A × A. It consists of ordered pairs (a, b) where a and b belong to set A.
What is the identity element for the operation of AND in Boolean algebra?
A 1
B A
C 0
D B
In Boolean algebra, the identity element for the AND operation is 1. This means that for any element A, A AND 1 = A. The identity element for OR is 0.
What is the result of (A ∩ B) ∪ (A ∩ C)?
A A ∩ B
B A ∪ (B ∩ C)
C A ∩ (B ∪ C)
D A ∩ B ∩ C
Using the distributive property of sets, the expression (A ∩ B) ∪ (A ∩ C) is equivalent to A ∩ (B ∪ C). This expression represents the set of elements in A that are also in either B or C.
In graph theory, what does a connected graph mean?
A Contains multiple edges
B Every pair of vertices is connected
C Contains only one vertex
D Contains no cycles
A connected graph is a graph where there is a path between every pair of vertices. This means that, starting from any vertex, you can reach any other vertex in the graph through edges.
Which of the following is true for a function f: A → B?
A Multiple elements in B map to one element in A
B A maps to multiple B
C Every element in B maps to one element in A
D Every element in A maps to one element in B
A function f: A → B means that each element in set A maps to exactly one element in set B. It defines a relation between elements of the two sets with no ambiguity or multiple mappings for any element in A.
What does the complement of a set A represent?
A Elements outside A
B A union B
C All elements of A
D Subset of A
The complement of a set A, denoted as A′, refers to all the elements that are in the universal set but not in A. It includes everything that is outside of A.
In Boolean algebra, what is the result of A AND NOT A?
A 0
B A
C NOT A
D 1
In Boolean algebra, A AND NOT A always results in 0. This is because A and NOT A are mutually exclusive; they cannot be true at the same time, so their AND operation results in false (0).
Which of the following graphs has no cycles?
A Complete graph
B Tree
C Directed graph
D Bipartite graph
A tree is a type of graph that has no cycles. It is a connected, acyclic graph where there is exactly one path between any two vertices, making it a special case of a connected graph.
What does the inverse of a relation mean?
A Reversing the graph
B Removing the relation
C Reversing set elements
D Swapping the order of pairs
The inverse of a relation R is formed by swapping the order of each pair in R. If R contains the pair (a, b), the inverse relation contains the pair (b, a).
What is the least common multiple (LCM) of 3 and 5?
A 15
B 30
C 5
D 10
The least common multiple (LCM) of two numbers is the smallest number that is divisible by both. For 3 and 5, the LCM is 15, as it is the smallest number that both 3 and 5 divide into without leaving a remainder.
What is the purpose of a truth table?
A To map functions
B To perform set operations
C To represent logical expressions
D To solve algebraic equations
A truth table is used to represent all possible truth values of a logical expression. It helps in analyzing and understanding the behavior of logical connectives like AND, OR, and NOT.
What is a matrix used for in Discrete Mathematics?
A Solving linear equations
B Representing graphs
C Calculating probabilities
D Representing sets
In Discrete Mathematics, matrices are commonly used to represent and solve systems of linear equations. Each row of a matrix corresponds to an equation, and the columns represent the coefficients of the variables.
What is the main feature of a directed graph?
A No vertices
B Multiple paths between vertices
C One direction per edge
D No edges
A directed graph (digraph) is characterized by edges that have a specific direction, indicated by an arrow. Each edge points from one vertex to another, signifying a one way relationship between the vertices.