Set Theory Basics Quiz

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Questions and Answers

Which of the following sets is an example of a finite set?

{x ∈ ℕ}

{1, 2, 3, 4, 5} (correct)

{1, 2, 3, 4, 5, ...}

{5} (correct)

What is the power set of A = {1, 2}?

{Φ, {1}, {2}, {1, 2}} (correct)

{1, 2}

{Φ, {1, 2}}

{1, 2, {1}, {2}}

Which of the following statements about the Cartesian product X × Y is true?

It is equal to the union of two sets.

It can only be performed on sets with the same number of elements.

It contains ordered pairs combining elements from both sets. (correct)

It results in a set that contains only single elements.

If A = {1, 2, 3} and B = {4, 5}, what is A ∩ B?

{} Signup and view all the answers

Which statement describes an antisymmetric relation?

If (a, b) and (b, a) are in R, then a must equal b. Signup and view all the answers

What is a defining feature of a universal relation?

It contains all possible pairs of a set with itself. Signup and view all the answers

What is the definition of a disjoint set?

Two sets that have no elements in common. Signup and view all the answers

Which set represents a singleton set?

{5} Signup and view all the answers

What must be true for a relation to be considered transitive?

If A→B and B→C, then A must be related to both B and C. Signup and view all the answers

Which set represents an equivalence relation?

{(1, 1), (2, 2), (3, 3)} Signup and view all the answers

Which of the following correctly describes a Hasse diagram?

It visually represents a partially ordered set. Signup and view all the answers

In a lattice, how is the join of two elements defined?

The smallest element that is greater than both elements. Signup and view all the answers

What is a characteristic of a totally ordered set?

All pairs of elements can be compared. Signup and view all the answers

What does the symmetric difference of two sets A and B represent?

Elements that are in either A or B, but not in both. Signup and view all the answers

What is the result of the operation A – B if A = {1, 2, 3} and B = {2}?

{1} Signup and view all the answers

Which statement correctly describes a multiset?

It contains duplicate elements and their counts. Signup and view all the answers

What is the result of the expression $A ∧ B$ when $A$ is false and $B$ is true?

False Signup and view all the answers

Given the implication $A → B$, when is the implication false?

When $A$ is true and $B$ is false Signup and view all the answers

What is the main characteristic of a bijective function?

Each element from the domain maps to exactly one unique element in the range. Signup and view all the answers

Which of the following represents a tautology?

(${A} → ${A})$ Signup and view all the answers

What type of logical statement is represented by the expression $(P₁ ∧ P₂ ∧ … ∧ Pₙ) → C$?

Argument Signup and view all the answers

What does the notation $nCk$ represent in combinatorics?

The number of ways to select items where order does not matter. Signup and view all the answers

Which of the following expresses the contrapositive of $A → B$?

~B → ~A Signup and view all the answers

Which statement is true regarding the sum rule in combinatorics?

The total number of outcomes is calculated by adding the individual outcomes of separate tasks. Signup and view all the answers

If a function $f(x) = 1 + 2x$ is given, what is the inverse function $f⁻¹(y)$?

f⁻¹(y) = (y - 1)/2 Signup and view all the answers

According to De Morgan's Laws, what is equivalent to $~(P ∧ Q)$?

~P ∨ ~Q Signup and view all the answers

Which of the following is the result of the expression $P ∨ F$?

Depends on P Signup and view all the answers

Which of these correctly describes the product rule in combinatorics?

The total number of outcomes is the product of the outcomes of two tasks performed sequentially. Signup and view all the answers

What is the result of applying double negation to $P$?

P Signup and view all the answers

How many different ways can you arrange 3 digits selected from the set {1, 2, 3, 4, 5} without repetition?

60 Signup and view all the answers

What is the outcome of the exclusive OR $A ⊕ B$ when both $A$ and $B$ are true?

False Signup and view all the answers

In a surjective function, what is true about the relationship between the domain and range?

Every element of the range is mapped by at least one element in the domain. Signup and view all the answers

Identify the type of expression represented by $P ∧ (P → Q)$.

Contingency Signup and view all the answers

What does it mean when a function is described as one-to-one?

Each element in the domain maps to a unique element in the range. Signup and view all the answers

What is the correct representation of the logical statement 'If I study from 5ME, then I am prepared for the exam'?

A → B Signup and view all the answers

How is the negation of 'All Engineers love 5ME' represented logically?

¬∀x [Love(x)] Signup and view all the answers

What is the outcome of the function composition g(f(x)) where f(x) = 2x + 1 and g(x) = x + 1?

2x + 3 Signup and view all the answers

Which statement correctly expresses the logical equivalence of 'Not all students are naughty'?

∃x [Student(x) ∧ ¬Naughty(x)] Signup and view all the answers

What is the definition of an injective function?

A function where no two elements in the domain map to the same element in the codomain. Signup and view all the answers

What is the correct formula to determine the number of injective functions from set A with 4 elements to set B with 4 elements?

4! Signup and view all the answers

In the expression ¬(∀x f(x)), what does this represent?

At least one x exists such that f(x) is false. Signup and view all the answers

How can 'Some Engineers love 5ME' be expressed in logical quantifiers?

∃x [Engineer(x) ∧ Love(x)] Signup and view all the answers

Study Notes

Basic Definitions

A set is a collection of well-defined objects or elements. Examples include {2, 4, 6, 8} and {1, 2, 3, 4, 5, ...∞}.
A singleton set has only one element, like {5}.
An empty or null set has no elements, represented as {} or Φ.
A finite set has a limited number of elements, while an infinite set has an unlimited number of elements.
A subset (S) is a set whose elements are all also elements of another set (A), denoted as S ⊆ A.
A power set (P(A)) contains all possible subsets of a set (A), including the empty set.
The number of elements in the power set is 2ⁿ, where 'n' represents the number of elements in the original set.
A universal set (U) contains all the elements being considered in a specific context.
Disjoint sets have no common elements, resulting in an empty intersection.

Basic Operations of a Set

Union (∪): Combines all elements from two sets.
- A ∪ B includes all elements that are in either A or B.
Intersection (∩): Identifies common elements between two sets.
- A ∩ B includes elements that are present in both A and B.
Difference (-): Removes elements from one set that are also present in a second set.
- A - B includes elements that are in A but not in B.
Cartesian Product (×): Creates a set of ordered pairs, combining elements from two sets.
- X × Y generates pairs where one element is from X and the other is from Y.

Relations

A relation is a subset of the Cartesian product of two sets.
The domain of a relation is the set of first elements in the ordered pairs.
The range of a relation is the set of second elements in the ordered pairs.
The total number of relations between two sets X and Y with 'm' and 'n' elements respectively, is 2^m+n.

Types of Relations

Empty relation: No relation exists between any elements in the sets.
Universal relation: Every element in set A is related to every element in set A.
Identity relation: Each element is related only to itself, forming a diagonal line when represented visually.
Inverse relation: Swaps the elements in each ordered pair of a relation.
Reflexive relation: All elements are related to themselves.
Symmetric relation: If (a, b) is in the relation, then (b, a) must also be in the relation.
Antisymmetric relation: If (a, b) and (b, a) are in the relation, then a must equal b.
Transitive relation: If (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.
Equivalence relation: A relation that is reflexive, symmetric, and transitive.
Partial order relation (POSET): A relation that is reflexive, antisymmetric, and transitive.
- A partially ordered set is denoted as [X, R], where X is the set and R is the partial order relation.

Hasse Diagram

A Hasse diagram visually represents a partial order relation.
It uses vertices to represent elements and edges to represent the relation.
Reflexive and transitive edges are omitted, and edges are oriented upwards.

Lattice

A lattice is a special Hasse diagram that satisfies the join and meet conditions.
- The join represents the least upper bound (UB) of two elements.
- The meet represents the greatest lower bound (LB) of two elements.

Finding Complement

The complement of an element in a lattice is another element that, when joined with the original element, results in the UB and when met with the original element, results in the LB.

Totally Ordered Set

A totally ordered set is a POSET where all elements are comparable to each other.

Venn Diagram

A Venn diagram visually represents sets using overlapping circles.
Each circle represents a set, and overlapping regions represent the intersection of sets.

Symmetric Difference

The symmetric difference of two sets contains elements that are in one set but not the other, and vice versa.

Difference

The difference of two sets contains elements that are in the first set but not in the second set.

Multiset

A multiset allows multiple copies of elements within a set.

Connectives (Operators)

NOT/ Negation (~): Reverses the truth value of a statement.
- True becomes False, and False becomes True.
Conjunction (∧) (and): True only when both statements are True.
Disjunction (∨) (or): True when at least one of the statements is True.
Implication (→): True unless the first statement is True and the second statement is False.
Exclusive Or (⊕): True only when one statement is True and the other is False.
Double Implication (↔): True only when both statements have the same truth value.

Converse, Inverse, and Contrapositive

Converse: Swaps the hypothesis and conclusion of an implication.
Inverse: Negates both the hypothesis and conclusion of an implication.
Contrapositive: Negates both the hypothesis and conclusion and swaps them.

Tautology, Contradiction, and Contingency

Tautology: A statement that is always True.
Contradiction: A statement that is always False.
Contingency: A statement that can be either True or False depending on the truth values of its components.

Propositional Equivalences

Two statements are propositionally equivalent if they have the same truth value for all possible combinations of truth values of their components.

Laws of Propositional Logic

Idempotence: P ∨ P = P and P ∧ P = P
Commutative: P ∨ q = q ∨ p and P ∧ q = q ∧ p
Associative: P ∨ (q ∨ r) = (p ∨ q) ∨ r and P ∧ (q ∧ r) = (p ∧ q) ∧ r
Distributive: P ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) and P ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)
Double negation: ~( ~p) = p
De Morgan's Laws: ~(p ∧ q) = ~p ∨ ~q and ~(p ∨ q) = ~p ∧ ~q

Identity Laws

P ∨ T = T
P ∨ F = P
P ∧ T = P
P ∧ F = F

Normal Form

Disjunctive Normal Form (DNF): A sum (OR) of products (AND) of literals.
Conjunctive Normal Form (CNF): A product (AND) of sums (OR) of literals.

Argument

An argument is a sequence of statements, called premises, that are intended to support a conclusion.

Modus Ponens

A valid argument form where the premises are P → q and P, and the conclusion is q.

Modus Tollen

A valid argument form where the premises are P → q and ¬q, and the conclusion is ¬p.

Hypothetical Syllogism

A valid argument form where the premises are P → q and q → r, and the conclusion is P → r.

Quantifiers

Universal quantifier (∀): Means "for all".
Existential quantifier (∃): Means "there exists".

Negation of Quantifiers

¬(∀x f(x)) = ∃x ¬f(x)
¬(∃x f(x)) = ∀x ¬f(x)

Functions

A function is a relation where each element in the domain is related to exactly one element in the range.
Function composition: f o g(x) = f(g(x)).

Injective Function (One-to-One)

An injective function maps distinct elements in the domain to distinct elements in the range.

Surjective Function (Onto)

A surjective function maps every element in the range to at least one element in the domain.

Bijective Function

A bijective function is both injective and surjective.

Inverse of a Function

The inverse of a function reverses the mapping, so if f(x) = y, then f⁻¹(y) = x.
The inverse function exists only if the original function is bijective.

Combinatorics

Combination: A selection of items where order does not matter.
Permutation: An arrangement of items where order matters.
Sum Rule: Used to calculate the number of choices when tasks are mutually exclusive.
Product Rule: Used to calculate the number of choices when tasks are sequential.

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Description

Test your knowledge on the fundamental concepts of set theory, including definitions of sets, types of sets, and basic operations. This quiz covers elements such as union, subsets, and power sets, ensuring a comprehensive understanding of the topic.