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Questions and Answers
What is the primary function of Boolean Algebra in computer science?
What is the primary function of Boolean Algebra in computer science?
Which of the following describes a property of a partially ordered set (poset)?
Which of the following describes a property of a partially ordered set (poset)?
In what way can lattices in posets be defined?
In what way can lattices in posets be defined?
Which application best illustrates the concept of mathematical induction?
Which application best illustrates the concept of mathematical induction?
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What does a well-formed formula in predicate logic require?
What does a well-formed formula in predicate logic require?
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Which of the following is NOT a rule of inference in predicate logic?
Which of the following is NOT a rule of inference in predicate logic?
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How does the concept of graph coloring relate to discrete structures?
How does the concept of graph coloring relate to discrete structures?
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What is a core disadvantage of using indirect proofs in mathematics?
What is a core disadvantage of using indirect proofs in mathematics?
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What defines an isomorphic ordered set?
What defines an isomorphic ordered set?
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Which of the following is a property of well-ordered sets?
Which of the following is a property of well-ordered sets?
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In a lattice, what does the term l.u.b. stand for?
In a lattice, what does the term l.u.b. stand for?
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What must be true about a subset for it to be considered well-ordered?
What must be true about a subset for it to be considered well-ordered?
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Which operation does not typically apply in lattice theory?
Which operation does not typically apply in lattice theory?
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What characteristic distinguishes a totally ordered set from a partially ordered set?
What characteristic distinguishes a totally ordered set from a partially ordered set?
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Which of the following can be an example of a partially ordered set?
Which of the following can be an example of a partially ordered set?
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What does the notation $a ∧ b$ represent in lattice theory?
What does the notation $a ∧ b$ represent in lattice theory?
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Which of the following properties describes the relationship where $a v a = a$?
Which of the following properties describes the relationship where $a v a = a$?
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What defines a bounded lattice?
What defines a bounded lattice?
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What is the condition for a lattice to be classified as distributive?
What is the condition for a lattice to be classified as distributive?
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In the context of the lattice P(S), which of the following statements is true?
In the context of the lattice P(S), which of the following statements is true?
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What is a characteristic of non-distributive lattices?
What is a characteristic of non-distributive lattices?
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Which operation is represented by the formula $a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)$?
Which operation is represented by the formula $a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)$?
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In a lattice, which element represents the least element?
In a lattice, which element represents the least element?
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Which of the following is an incorrect statement regarding the absorption properties?
Which of the following is an incorrect statement regarding the absorption properties?
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Study Notes
Course Information
- Course: Discrete Structures & Theory of Logic
- Unit: 3
- Course Code: BCSE0306
- Instructor: M. Abdul Mateen Siddiqui
- Semester: 3rd
- Date: 11/05/2020
Unit 3 Objectives
- State the algebraic definition of a Boolean algebra
- The objective of unit 3 to solve problems using the algebraic properties of the elements of a Boolean algebra
- Define a poset and find the maximum and minimum elements of subsets of posets when they exist
- Find the supremum and infimum of subsets of posets when they exist
- Define a lattice and identify lattices among posets
Content
- Video links
- Daily Quiz
- Weekly Assignment
- MCQ
- Old Question papers
- Expected Question for University Exam
- Summary
- References
Topic Prerequisites & Recap (CO3)
- Basic understanding of class 10 mathematics NCERT
- Basic knowledge of sets and algebraic rules
- Basic understanding of set theory, relations, and functions
- Basic understanding of algebraic structures
Topic Objectives: Lattices and Boolean Algebra (CO3)
- Represent a Hasse Diagram
- Give examples of finite and infinite sets
- Build new sets from existing sets by applying various combinations of set operations (intersection, union, difference, complement)
- Determine Posets
- Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets.
Topic: Ordered Sets & Posets
- Partial ordering: reflexive, anti-symmetric, and transitive binary relation
- Example: For S = {a, b, c}, partial ordering satisfies
- Reflexive: a ≤ a
- Anti-Symmetric: a ≤ b and b ≤ a imply a = b
- Transitive: if a ≤ b and b ≤ c, then a ≤ c
Topic: Terminologies
- Cartesian Product: the product of two sets A and B such that every element of set A relates to every other element of set B to form ordered pairs. A*B = {(a,1), (a,2), (b,1), (b,2)}
- Reflexive Relation: A relation R, over a set A, is reflexive if every element of the set is 'related' to itself.
- Symmetric: A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R, for all a, b ∈ A.
- Anti-symmetric Relation: A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b, is called anti-symmetric
- Transitive: A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A.
Topic: Hasse Diagram (CO3)
- Pictorial representation of POSET is called Hasse Diagram.
- Steps:
- Draw Diagraphs
- Remove reflexive and transitive edges
- Draw diagraph in such a way that all edges are pointing upwards
- Remove directions and replace circle with dot
Topic: Upper and Lower Bound
- Upper Bound: An element x ∈ A is called an upper bound of B if (y, x) ∈ poset for every y ∈ B
- Lower Bound: An element y ∈ A is called a lower bound of B if (x, y) ∈ poset for every x ∈ B
Topic: Least Upper Bound (Supremum) & Greatest Lower Bound (Infimum)
- Least Upper Bound(SUPREMUM): An element M in S is called an upper bound of A if M succeeds every element of A. If an upper bound of A precedes every other upper bound of A, then it's the supremum of A, denoted by Sup (A).
- Greatest Lower Bound (INFIMUM): An element m in S is called a lower bound of A if m precedes every element of A. If a lower bound of A succeeds every other lower bound of A, then it's the infimum of A, denoted by Inf (A).
Topic: Comparable and Non-Comparable Elements
- Comparable Elements: Two elements a and b of a set A are comparable if a≤b or b≤a.
- Non-comparable Elements: a and b are not comparable elements if neither a≤b nor b≤a.
Topic : Isomorphic Ordered Sets
- If (A, ≤) and (B, ≤) are two partially ordered sets, they are isomorphic if their structures are entirely similar.
Topic: Well Ordered Set
- A partially ordered set is called a well-ordered set if every non-empty subset has a least element.
- Example: The natural numbers with the "less than or equal to" relation
Topic: Lattice
- A lattice is a poset in which every pair of elements has a least upper bound (join) and a greatest lower bound (meet).
Topic: Properties of Lattices
- Idempotent Properties:
- av a = a
- a∧ a = a
- Commutative Properties:
- av b = b v a
- a∧ b = b∧ a
- Associative Properties:
- a v (b v c) = (a v b) v c
- a ∧ (b ∧ c) = (a ∧ b) ∧ c
- Absorption Properties:
- a v (a Λ b) = a
- a Λ (a v b) = a
Topic: Bounded Lattices
- A lattice L is said to be bounded if it has a greatest element 1 and a least element 0
Topic: Distributive Lattices
- A lattice (L, ≤) is called distributive if for any elements a, b, and c in L, we have:
- a ∧ (b v c) = (a ∧ b) v (a ∧ c)
- a v (b ∧ c) = (a v b) ∧ (a v c)
Topic: Complemented Lattices
- A lattice L is said to be complemented if it is bounded and every element in it has a complement.
Course Objectives
- Apply the basic principles of sets, relations, and functions.
- Understand algebraic structures and their properties.
- Describe lattices and apply Boolean algebra.
- Infer the validity of statements using predicate logic.
- Design and use non-linear data structures like trees and graphs.
Program Outcomes (PEOs)
- PEO 1: Excellent scientific & engineering breadth to design sustainable solutions for real-life problems.
- PEO 2: Successful career in industries including higher studies or entrepreneurial endeavors.
- PEO 3: Effective communication skills, professional attitudes, and desire to learn knowledge in emerging trends.
- PEO 4: Life-long learning.
Program Specific Outcomes (PSOs)
- PSO 1: Identify, analyze & design ethical solutions using AI, robotics, technology.
- PSO 2: Design & develop hardware devices and interfacing software.
- PSO 3: Understand interdisciplinary computing techniques and apply them in design of advanced computing.
- PSO 4: Conduct investigations of complex problems with technical, managerial, leadership, and modern engineering tools provided by industry.
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