Download to read offline
More Related Content
Similar to Theory of computation, relation, function, mathematical induction
![Presentation_by_HADI[2][1] [Read-Only].pptx](https://cdn.slidesharecdn.com/ss_thumbnails/presentationbyhadi21read-only-250129135223-a3179f8f-thumbnail.jpg?width=640&height=640&fit=bounds)
Theory of computation, relation, function, mathematical induction
- 1. Code: BCEECE311 Theory ofComputation Dr. Bharti Salunke
- 2. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 2 Detailed Syllabus
- 3. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 3
- 4. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 4 Need of TOC • TOC provides the mathematical basis for understanding how computation works, defining what problems can or cannot be solved by machines. • It helps in designing abstract machines (like Finite Automata, Pushdown Automata, Turing Machines) to model and solve real-world problems systematically. • Essential for building compilers, interpreters, and parsers, as it classifies languages into Regular, Context-Free, and others—each with specific machine models. • TOC helps in understanding time and space complexity.
- 5. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 5 Application of TOC • Compiler Design: TOC concepts like finite automata and grammars are used in lexical analysis and syntax parsing in compilers. • Artificial Intelligence: State machines help model decision-making processes in AI agents and robotics. • Natural Language Processing (NLP): Regular and context-free grammars are used for syntax checking, sentence parsing, and Chatbot design. • Cryptography and Security: Computability and complexity theory guide the design of secure encryption and authentication algorithms. • Software Verification and Model Checking: TOC helps ensure that software and hardware systems behave correctly and reliably through formal verification methods.
- 6. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 6 Basic of Sets
- 7. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 7
- 8. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 8
- 9. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 9 Type Example Finite Set A = {1, 2, 3, 4} Infinite Set N = {1, 2, 3, …} Empty Set ( ) ∅ B = {} Subset A B every element of A is in B ⊆ ⇒ Power Set P(A) = set of all subsets of A Universal Set U = set containing all elements under consideration Equal Sets A = {1, 2}, B = {2, 1} A = B ⇒ Types of Sets
- 10. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 10 Operation Symbol Description Union A B ∪ All elements in A or B or both Intersection A ∩ B Common elements in A and B Difference A − B Elements in A but not in B Complement A' Elements not in A (in Universal Set) Cartesian Product A × B Set of ordered pairs (a, b) with a ∈ A, b B ∈ Set Operations
- 11. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 11 Set Operations Union ( ), Intersection ( ∩ ), Difference ( − ) ∪ • A = {1, 2, 3}, B = {3, 4, 5} A B = {1, 2, 3, 4, 5} ∪ A ∩ B = {3} A − B = {1, 2} Complement ( A' or U − A ) • If U = {1, 2, 3, 4, 5} and A = {2, 3} Then A = {1, 4, 5} ′
- 12. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 12 Set Operations Symmetric Difference ( or ) ⊕ △ (A − B) (B − A) ∪ • A = {1, 2, 3}, B = {3, 4, 5} A B = {1, 2, 4, 5} △ Cartesian Product ( × ) • A = {1, 2}, B = {x, y} A × B = {(1, x), (1, y), (2, x), (2, y)}
- 13. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 13 Questions 1. Let A = {a, b, c}, B = {b, c, d, e}. Find A B. ∪ 2. A = {1, 3, 5, 7}, B = {2, 3, 5, 8}. Find A ∩ B. 3. A = {10, 20, 30, 40}, B = {20, 50, 60}. Find A − B. 4. A = {p, q, r}, B = {q, r, s, t}. Find A B. △ 5. Universal set U = {1, 2, 3, 4, 5, 6}, Set A = {2, 4, 6}. Find A . ′ 6. A = {1, 2}, B = {x, y}. Find A × B.
- 14. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 14 Questions Let A = {a, b, c}, B = {b, c, d, e}. Find A B. ∪ A B = {a, b, c, d, e} ∪ A = {1, 3, 5, 7}, B = {2, 3, 5, 8}. Find A ∩ B. A ∩ B = {3, 5} A = {10, 20, 30, 40}, B = {20, 50, 60}. Find A − B. A − B = {10, 30, 40}
- 15. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 15 Questions Universal set U = {1, 2, 3, 4, 5, 6}, Set A = {2, 4, 6}. Find A . ′ A = {1, 3, 5} ′ A = {p, q, r}, B = {q, r, s, t}. Find A B. △ A B = {p, s, t} △ A = {1, 2}, B = {x, y}. Find A × B. A × B = {(1, x), (1, y), (2, x), (2, y)}
- 16. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 16 Questions • Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, U = {1, 2, 3, 4, 5, 6, 7} Find: • A ∩ B = ? • A B = ? ∪ • B − A = ? • A = ? ′
- 17. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 17 Answers • Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, U = {1, 2, 3, 4, 5, 6, 7} Answers: A ∩ B = {3, 4} A B = {1, 2, 3, 4, 5, 6} ∪ B − A = {5, 6} A = {5, 6, 7} ′
- 18. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 18 Relations • A relation is a subset of the Cartesian product of two sets. • If you have sets A and B, a relation R from A to B is any subset of A×B. Example 1: Basic Relation Let: • A={1,2} • B={x,y} Then A×B={(1,x),(1,y),(2,x),(2,y)} A relation R A×B could be: ⊆ R = { (1, x), (2, y) }
- 19. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 19 Relations
- 20. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 20 Relations
- 21. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 21 Relations
- 22. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 22 Types of Relation 1. Reflexive Relation • Definition: A relation R on set A is reflexive if every element is related to itself. • Condition: a A, (a, a) R ∀ ∈ ∈ • Example: On A = {1, 2, 3}, R = {(1,1), (2,2), (3,3)} 2. Irreflexive Relation • ∀a A,(a,a) R ∈ ∉ • No self-pairs like (1,1)
- 23. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 23 Types of Relation 3. Symmetric Relation • Definition: A relation R is symmetric if (a, b) R implies (b, a) R. ∈ ∈ • Example: If (1, 2) R, then (2, 1) must also be in R. ∈ 4. Anti-Symmetric Relation • Definition: A relation R is anti-symmetric if (a, b) R and (b, a) R ∈ ∈ implies a = b. • Example: On A = {1, 2}, R = {(1,1), (2,2), (1,2)} is anti-symmetric because (2,1) is not in R
- 24. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 24 Types of Relation 5. Asymmetric Relation In an asymmetric relation, if the pair (a, b) is in the relation, then the pair (b, a) must not be in the relation for any elements a and b from the set. ∀ a, b A ∈ , if (a, b) R ∈ then (b, a) R ∉ and vice versa. 6. Transitive Relation • Definition: A relation R is transitive if (a, b) R and (b, c) R implies ∈ ∈ (a, c) R. ∈ • Example: If (1,2) R and (2,3) R, then (1,3) must also be in R. ∈ ∈
- 25. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 25 Types of Relation 6. Equivalence Relation • Definition: A relation that is reflexive, symmetric, and transitive. • Example: "Is congruent modulo n" on integers. a≡b (mod n) a−b=kn, for some integer k. 17 ≡ 5 (mod 12) Because 17−5=12, and 12 is divisible by 12. 7. Partial Order Relation (Poset) • Definition: A relation that is reflexive, anti-symmetric, and transitive. • Example: “≤” on set of natural numbers.
- 26. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 26 Example Let A={1,2,3} Define relation R={(1,1),(2,2),(3,3),(1,2),(2,1)} Let's test whether it's an equivalence relation: • Reflexive? Yes, because all elements of A appear as (a,a): (1,1), (2,2), (3,3) • Symmetric? (1,2) is in R, and (2,1) is also in R→ OK • Transitive? (1,2) and (2,1) are in R, but (1,1) is already in R, so OK. No other pairs violate transitivity.
- 27. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 27
- 28. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 28
- 29. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 29
- 30. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 30 Closure of Relations • In mathematics, especially in the context of set theory and algebra, the closure of relations is a crucial concept. • It involves extending a given relation to include additional elements based on specific properties, such as reflexivity, symmetry, and transitivity. 1. Reflexive Closure • The reflexive closure of a relation R on a set A is the smallest relation R that ′ contains R and is reflexive. This means that every element in A is related to itself. • Formula: R = R {(a,a) a A} ′ ∪ ∣ ∈ • Example: Let A = {1,2} and R={(1,2)}. The reflexive closure of R is: R = {(1,2),(1,1), ′ (2,2)}.
- 31. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 31 2. Symmetric Closure • The symmetric closure of a relation R on a set A is the smallest relation R that contains R and is symmetric. ′ • Formula: R = R {(b,a) (a,b) R} ′ ∪ ∣ ∈ • Example: Let A = {1,2} and R = {(1,2)}. The symmetric closure of R is: R = {(1,2),(2,1)}. ′
- 32. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 32 3. Transitive Closure • The transitive closure of a relation R on a set A is the smallest relation R that contains R and is transitive. ′ Example: • Let A = {1,2,3} and R = {(1,2),(2,3)}. The transitive closure of R is: • R = {(1,2),(2,3),(1,3)}. ′
- 33. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 33 Mathematical Induction • Mathematical induction is a proof technique used to prove that a statement is true for all natural numbers (usually starting from 1 or 0). Steps of Mathematical Induction • To prove a statement P(n) true for all n≥n0 , follow these steps: Step 1: Base Case • Prove that the statement is true for the initial value (usually n=1 or n=0). Step 2: Inductive Hypothesis • Assume that the statement is true for n=k. i.e., Assume P(k) is true.
- 34. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 34 Step 3: Inductive Step • Prove that if the statement is true for n=k, then it must also be true for n=k+1. i.e., show that P(k) P(k+1) ⇒
- 35. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 35 Prove that
- 36. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 36
- 37. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 37
- 38. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 38
- 39. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 39
- 40. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 40
- 41. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 41 Solution 3:
- 42. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 42
- 43. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 43 Solution 4:
- 44. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 44 Solution 5:
- 45. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 45 Example 6:
- 46. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 46
- 47. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 47
- 48. Department of ComputerScience And Engineering, Poornima University, Jaipur, India 48
Editor's Notes
- #6 National Institute of Standards and Technology, US
- #7 National Institute of Standards and Technology, US
- #8 National Institute of Standards and Technology, US
- #9 National Institute of Standards and Technology, US
- #10 National Institute of Standards and Technology, US
- #11 National Institute of Standards and Technology, US
- #12 National Institute of Standards and Technology, US
- #13 National Institute of Standards and Technology, US
- #14 National Institute of Standards and Technology, US
- #15 National Institute of Standards and Technology, US
- #16 National Institute of Standards and Technology, US
- #17 National Institute of Standards and Technology, US
- #18 National Institute of Standards and Technology, US
- #19 National Institute of Standards and Technology, US
- #20 National Institute of Standards and Technology, US
- #21 National Institute of Standards and Technology, US
- #22 National Institute of Standards and Technology, US
- #23 National Institute of Standards and Technology, US
- #24 National Institute of Standards and Technology, US
- #25 National Institute of Standards and Technology, US