![Set theory
● (d1) A∩B = df {x: x∈A & x∈B} [simple intersection]
● (d2) A–B = df {x: x∈A & x≠B} [set-difference]
● (d3) A∪B = df {x: x∈A ∨ x∈B} [simple union]
● (d4) •(f) = df {x: ∃Y(Y∈f & x∈Y)} [general union]
● (d5) €(f) = df {x: ∀Y(Y∈f → x∈Y)} [general intersection]](https://image.slidesharecdn.com/tocday1-200618050300/85/Theory-of-Computation-Introduction-Session-9-320.jpg)
Theory of Computation Introduction Session
- 1. Theory of Computation By Rushabh Wadkar
- 2. Topics to be covered Introduction to TOC i. Solvability of a problem ii. Order of an algorithm(asymptotic notations) iii. Set theory iv. Graphs and trees Formal languages Day 1
- 3. Introduction to TOC In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The theory of computation can be considered the creation of models of all kinds in the field of computer science. Therefore, mathematics and logic are used.
- 4. Solvability of a problem Solvable Problems ● The problem has a definite solution. ● The problem will be solved in finite number of steps. ● Example: Shortest Path problems Unsolvable Problems ● The problem has no definite solution. ● The solution doesn’t exist yet. No finite steps can provide u with a solution. ● Example: Division by Zero All problems are divided into 2 categories, Solvable and Unsolvable
- 5. Königsberg bridge problem The Königsberg bridge problem asks if the seven bridges of the city of Königsberg (formerly in Germany but now known as Kaliningrad and part of Russia) over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began.
- 6. Königsberg bridge problem Let us consider each land like a node of a graph. And the bridges are the edges connecting the nodes. The degree of each node is odd, hence it is not possible to start from a particular land and come back there without traversing a bridge more than once.
- 7. PROBLEM SOLVABLE UNSOLVABLE DECIDABLE UNDECIDABLE Can be moved to ● Algorithm + Procedure exists ● Only Procedure exists
- 8. Asymptotic Notations ● f(n)=O( g(n)) "Big O(micron)" – upper bound => worst case ● f(n) = Ω(g(n)) "Big Omega" – lower bound => best case ● f(n) = θ(g(n)) "Big Theta" – upper & lower bound => "average" case
- 9. Set theory ● (d1) A∩B = df {x: x∈A & x∈B} [simple intersection] ● (d2) A–B = df {x: x∈A & x≠B} [set-difference] ● (d3) A∪B = df {x: x∈A ∨ x∈B} [simple union] ● (d4) •(f) = df {x: ∃Y(Y∈f & x∈Y)} [general union] ● (d5) €(f) = df {x: ∀Y(Y∈f → x∈Y)} [general intersection]
- 10. Relations ● R is a relation => R is a set of ordered pair (a,b)∈AxB ● R is a relation from A to B iff it satisfies the following restrictions dom(R) ⊆ A ran(R) ⊆ B
- 11. Functions ● A function is, by definition, a relation R satisfying the following restriction. ∀xyz(xRy & xRz .→ y=z) No two images will have same preimage.
- 12. DeMorgan’s Law ● (A∪B)’= A’∩ B’ ● (A∩B)’= A’∪ B’
- 13. Important Set Properties ● Disjoint Sets: (A∩B) = ∅ ● Size of Set= |S|, No. of elements in a set
- 14. ● Graph consists of edges and nodes. ● A non-cyclic graph is called a tree. ● There exist directed and undirected graphs.
- 15. Formal Languages & its comparison Language: English Formal Language (For a fan) Symbols: (A . . . Z) (a . . . z) 0 & 1 On and off states Alphabet: Set of all symbols (A, B . . . Y, Z, a, b . . . y, z) Set of all symbols (0 , 1) Strings: Any set of words (including, these, themselves) (0 , 1, 01, 10, 001, . . . )
- 16. End of Day 1 www.linkedin.com/in/wadkar-rushabh @RushabhWadkar Thank you...