A set is defined as a collection of distinct objects or elements. These elements can be anything, such as numbers, letters, or other objects. The order of elements in a set doesn’t matter, and each element appears only once.
What does a Venn diagram represent?
A Set relationships
B Logical operations
C Graphical data
D Mathematical equations
Venn diagrams are used to represent set relationships. They visually depict the inclusion or exclusion of elements within sets and can show operations such as union, intersection, and complement.
Which of the following is the correct symbol for the intersection of two sets?
A ⊂
B ∪
C ⊄
D ∩
The symbol “∩” represents the intersection of two sets, which includes all elements that are common to both sets. The union (∪) represents all elements from both sets, and the subset symbols (⊂, ⊄) represent the inclusion of one set in another.
Which logic gate produces a true output only when all inputs are true?
A NOT
B XOR
C AND
D OR
The AND gate outputs true (1) only when all of its inputs are true (1). If any of the inputs are false (0), the output will be false (0). The OR gate outputs true if at least one input is true, while XOR produces true if exactly one input is true.
What is the purpose of a truth table?
A To perform arithmetic operations
B To represent logical functions
C To solve equations
D To find set relations
A truth table is a mathematical table used to represent the output of a logical function for all possible input values. It is commonly used in logic to analyze and understand the behavior of logical expressions and gates.
In a directed graph, what do the edges represent?
A Points in space
B Relationships in sets
C Connections between nodes
D Pathways of logic
In a directed graph, edges represent the connections between nodes (vertices). Each edge has a direction, indicated by an arrow, which shows the direction of the relationship between the nodes.
What is the degree of a node in a graph?
A Number of edges attached to it
B Distance between nodes
C Number of vertices it connects to
D Number of paths available
The degree of a node in a graph is the number of edges that are incident to it. For a directed graph, it can be further classified into in degree and out degree, depending on the direction of the edges.
Which of the following is true for a function?
A A function maps elements randomly
B One element can map to multiple outputs
C A function doesn’t require any inputs
D Multiple elements can map to one output
A function maps each input (domain) to a single output (codomain), but multiple inputs can map to the same output. This is known as many to one mapping. A one to one mapping would require that each input corresponds to a unique output.
What is the result of the following Boolean expression: A AND (B OR C)?
A (A AND B) OR C
B A OR (B AND C)
C A AND (B OR C)
D A AND B AND C
The expression “A AND (B OR C)” means that the result is true only when A is true and at least one of B or C is true. This is a straightforward interpretation of the AND and OR operations in Boolean algebra.
What is the basic unit of a Boolean algebra?
A Variable
B Operand
C Set
D Truth table
The basic unit of Boolean algebra is a variable, which can take values of 0 or 1 (false or true). Boolean algebra operates on variables using logical operations such as AND, OR, and NOT.
Which of the following is a recurrence relation?
A a = b + c
B a(n) = a(n-1) + 1
C y = mx + b
D x = x + 1
A recurrence relation is an equation that defines the value of a function in terms of its previous values. The relation “a(n) = a(n-1) + 1” expresses the value of a(n) based on its previous value, a(n-1).
What is the purpose of a base case in mathematical induction?
A To validate the last step
B To prove that the statement holds for all numbers
C To prove the hypothesis
D To validate the first step
The base case in mathematical induction is the initial step that proves the statement is true for the first value of the variable, usually 1. This step is crucial before proceeding to prove the general case.
What does a lattice in mathematics represent?
A A grid of points
B A geometric figure
C A set with certain operations
D A set of numbers
A lattice is a partially ordered set in which every two elements have a unique least upper bound (supremum) and greatest lower bound (infimum). Lattices are used in various branches of mathematics, including Boolean algebra and set theory.
What does modular arithmetic deal with?
A Remainders after division
B Numbers in sequences
C Solving quadratic equations
D Operations on functions
Modular arithmetic focuses on the remainders that result from dividing numbers. For example, in mod 5, the numbers 7 and 2 are congruent because both leave a remainder of 2 when divided by 5.
What is a graph that can be drawn without any edges crossing called?
A Bipartite graph
B Planar graph
C Tree
D Directed graph
A planar graph is one that can be drawn in a plane without any edges crossing. This property is important in various areas of graph theory, particularly in the study of graph embedding and topology.