Exploring Sets and Logic

What is the result of the union of sets A = {1, 2, 3} and B = {3, 4, 5}?

• {1, 2, 3}

• {1, 2, 3, 4, 5}

• {1, 2, 4, 5}

• {3, 4, 5}

If A = {x | x is an even number} and B = {x | x is a prime number}, what is A ∩ B?

• {4}

• {6}

• {3}

• {2}

Define a multiset and explain how it differs from a set.

• A multiset is a collection that cannot have duplicates.

• A multiset can only contain unique elements.

• A multiset is a set with no elements.

• A multiset is a generalized concept of a set that allows for multiple occurrences of the same element. Unlike a set, where each element is unique, a multiset can contain duplicates.

What is the cardinality of the multiset M = {a, a, b, c}?

• 4

• 3

• 6

• 5

Explain the concept of set difference and provide an example.

• Set difference is the same as union.

• If A = {1, 2} and B = {1, 2}, then A \ B = {}

• For example, if A = {1, 2, 3} and B = {2, 3, 4}, then the set difference A \ B = {1}.

• A \ B = {2, 3, 4}

In logic, what is a tautology? Give an example.

• A statement that is sometimes true and sometimes false.

• A statement that is always false regardless of the truth values.

• A statement that can only be true if A is true.

• An example of a tautology is the statement 'A or not A', which is always true regardless of whether A is true or false.

How can multisets be used in database theory?

• Multisets are used to enforce unique constraints in databases.

• Multisets allow for the representation of collections with duplicate elements, useful in scenarios like inventory management and aggregate query optimization.

• Multisets eliminate the need for indexing in database systems.

• Multisets are primarily used for data encryption in databases.

What is the power set of the set S = {x, y}?

• { {}, {x}, {y}, {x, y} }

• { {}, {x} }

• { {y}, {x, y} }

• { {x, y}, {x, y} }

If P and Q are two propositions, what does the expression P ∧ Q represent?

• P → Q represents 'If P then Q'.

• P ∧ Q represents 'P and Q'.

• ¬P represents 'Not P'

• P ∨ Q represents 'P or Q'.

Describe a real-world application of set operations in computer science.

• Using set operations for image processing in graphics software.

• Database query optimization and data retrieval using SQL operations like UNION.

• Implementing set operations in machine learning algorithms for classification.

• Applying set operations in network routing protocols for data transmission.