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Last Minute Notes – Discrete Mathematics

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Propositional Logic

  1. Implication( →): For any two propositions p and q, the statement “if p then q” is called an implication and it is denoted by p → q.
  2.  \begin{tabular}{ |c|c|c| }      \hline     p & q & p\rightarrow q\\     \hline     \hline     T & T & T\\     \hline     T & F & F\\     \hline     F & T & T\\     \hline     F & F & T\\     \hline     \end{tabular}
  3. if and only if(↔): For any two propositions p and q, the statement “p if and only if(iff) q” is called a biconditional and it is denoted by p ↔ q.
  4.  \begin{tabular}{ |c|c|c| }      \hline     p & q & p\leftrightarrow q\\     \hline     \hline     T & T & T\\     \hline     T & F & F\\     \hline     F & T & F\\     \hline     F & F & T\\     \hline     \end{tabular} De Morgan’s Law :
    •  \neg (p\wedge q) \equiv \neg p \vee \neg q
    •  \neg (p\vee q) \equiv \neg p \wedge \neg q
      Special Conditional Statements 1.Implication : p\rightarrow q 2.Converse : The converse of the proposition p\rightarrow q is q\rightarrow p 3.Contrapositive : The contrapositive of the proposition p\rightarrow q is \neg q\rightarrow \neg p 4.Inverse : The inverse of the proposition p\rightarrow q is \neg p\rightarrow \neg q Types of propositions based on Truth values 1.Tautology – A proposition which is always true, is called a tautology. 2.Contradiction – A proposition which is always false, is called a contradiction. 3.Contingency – A proposition that is neither a tautology nor a contradiction is called a contingency. There are two very important equivalences involving quantifiers
    1.  \forall x(P(x)\wedge Q(x)) \equiv \forall xP(x) \wedge \forall xQ(x)
    2.  \exists x(P(x)\vee Q(x)) \equiv \exists xP(x) \vee \exists xQ(x) 
    Rules of inference   \begin{tabular}{||c||c||c||} \hline Rule of Inference & Tautology & Name\\ \hline  \rule{0pt}{8ex} \shortstack[l]{p \\ p\rightarrow q \\ \rule{1cm}{0.5pt}\\ \therefore q}& (p\wedge (p\rightarrow q)) \rightarrow q & Modus Ponens \\ \hline  \rule{0pt}{8ex} \shortstack[l]{\neg q \\ p\rightarrow q \\ \rule{1cm}{0.5pt}\\ \therefore \neg p}& (\neg q \wedge (p\rightarrow q)) \rightarrow \neg p & Modus Tollens \\ \hline  \rule{0pt}{8ex} \shortstack[l]{p\rightarrow q \\ q\rightarrow r \\ \rule{1.3cm}{0.5pt}\\ \therefore p \rightarrow r}& ((p\rightarrow q) \wedge (q\rightarrow r)) \rightarrow (p\rightarrow r) & Hypothetical syllogism \\ \hline  \rule{0pt}{8ex} \shortstack[l]{ \neg p \\ p\vee q \\ \rule{0.8cm}{0.5pt}\\ \therefore q} & (\neg p \wedge (p\vee q)) \rightarrow q & Disjunctive Syllogism \\ \hline  \rule{0pt}{8ex} \shortstack[l]{p \\ \rule{1.5cm}{0.5pt} \\ \therefore (p \vee q)}& p\rightarrow (p\vee q) & Addition \\ \hline  \rule{0pt}{8ex} \shortstack[l]{ (p\wedge q)\rightarrow r \\ \rule{2.3cm}{0.5pt}\\ \therefore p\rightarrow (q\rightarrow r)} & ((p\wedge q)\rightarrow r) \rightarrow (p\rightarrow (q\rightarrow r)) & Exportation\\ \hline  \rule{0pt}{8ex} \shortstack[l]{p\vee q\\\neg p\vee r \\ \rule{1.2cm}{0.5pt} \\ \therefore q\vee r}& ((p\vee q) \wedge(\neg p\vee r)) \rightarrow q\vee r & Resolution \\ \hline   \end{tabular}


    Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects.

         \begin{flalign*} P(n, r) &= n * (n-1) * ... * (n-r+1)\\ &= \frac{n * (n-1) * ... * (n-r+1) * (n-r) *...* 2 * 1}{(n-r) * (n-r-1) * .... * 2 * 1}\\ &= \frac{n!}{(n-r)!} \end{flalign*}

    Combination: A combination of a set of distinct objects is just a count of the number of ways a specific number of elements can be selected from a set of a certain size. The order of elements does not matter in a combination.  \therefore P(n, r) = C(n, r) * P(r, r)\\ which gives us-

         \begin{flalign*} C(n, r) &= \frac{P(n, r)}{P(r, r)}\\ &= \frac{n!}{(n-r)!} * \frac{1}{r!}\\ &= \frac{n!}{r!(n-r)!}& \end{flalign*}

    Binomial Coefficients: The r-combinations from a set of n elements if denoted by \binom{n}{r}. This number is also called a binomial coefficient since it occurs as a coefficient in the expansion of powers of binomial expressions. Let x and y be variables and n be a non-negative integer. Then

         \begin{flalign*} (x+y)^n &= \sum_{j=0}^{n} \binom{n}{j} x^{n-j}y^j\\ &= \binom{n}{0}x^{n} + \binom{n}{1}x^{n-1}y +...+ \binom{n}{n-1}xy^{n-1} + \binom{n}{n}y^{n} \end{flalign*}

    The binomial expansion using Combinatorial symbols
    (a+b)^n = ^nC_0 a^n b^0 + ^nC_1 a^{n-1} b^1 + ^nC_2 a^{n-2} b^2 .. + ^nC_{n-k} a^k b^{n-k} .. +^nC_n a^0 b^n

    Set Theory

    A Set is an unordered collection of objects, known as elements or members of the set. An element ‘a’ belong to a set A can be written as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A. Equal sets Two sets are said to be equal if both have same elements. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets. NOTE: Order of elements of a set doesn’t matter. Subset A set A is said to be subset of another set B if and only if every element of set A is also a part of other set B. Denoted by ‘‘. ‘A ⊆ B ‘ denotes A is a subset of B. To prove A is the subset of B, we need to simply show that if x belongs to A then x also belongs to B. To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B. asubsetB ‘U’ denotes the universal set. Above Venn Diagram shows that A is a subset of B. Size of a Set Size of a set can be finite or infinite. For example
    Finite set: Set of natural numbers less than 100.
    Infinite set: Set of real numbers.
    Size of the set S is known as Cardinality number, denoted as |S|. Note: Cardinality of a null set is 0. Power Sets The power set is the set all possible subset of the set S. Denoted by P(S). Example: What is the power set of {0, 1, 2}? Solution: All possible subsets {∅}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}. Note: Empty set and set itself is also the member of this set of subsets. Cardinality of power set is 2^n, where n is the number of elements in a set. Cartesian Products Let A and B be two sets. Cartesian product of A and B is denoted by A × B, is the set of all ordered pairs (a, b), where a belong to A and b belong to B.
    A × B = {(a, b) | a ∈ A ∧ b ∈ B}.
    The cardinality of A × B is N*M, where N is the Cardinality of A and M is the cardinality of B. Note: A × B is not the same as B × A. Union Union of the sets A and B, denoted by A ∪ B, is the set of distinct element belongs to set A or set B, or both. Intersection The intersection of the sets A and B, denoted by A ∩ B, is the set of elements belongs to both A and B i.e. set of the common element in A and B. Disjoint Two sets are said to be disjoint if their intersection is the empty set .i.e sets have no common elements. Set Difference Difference between sets is denoted by ‘A – B’, is the set containing elements of set A but not in B. i.e all elements of A except the element of B. Complement The complement of a set A, denoted by A^\complement, is the set of all the elements except A. Complement of the set A is U – A.
    1. A\cup B =n(A) + n(B) - n(A\cap B)

    2. A-B=A\cap \bar{B}

    Group A non-empty set G, (G, *) is called a group if it follows the following axiom:
    • Closure:(a*b) belongs to G for all a, b ∈ G.
    • Associativity: a*(b*c) = (a*b)*c ∀ a, b, c belongs to G.
    • Identity Element:There exists e ∈ G such that a*e = e*a = a ∀ a ∈ G
    • Inverses:∀ a ∈ G there exists a-1 ∈ G such that a*a-1 = a-1*a = e

    Relations And Functions

    |A| = m and |B| = n, then 1. No. of functions from A to B = nm
    2. No. of one to one function = (n, P, m)
    3. No. of onto function =nm – (n, C, 1)*(n-1)m + (n, C, 2)*(n-2)m …. +(-1)m*(n, C, n-1), if m >= n; 0 otherwise
    4. Necessary condition for bijective function |A| = |B|
    5. The no. of bijection function =n!
    6. No. of relations =2mn
    7. No. of reflexive relations =2n(n-1)
    8. No. of symmetric relations = 2n(n+1)/2
    9. No. of Anti Symmetric Relations = 2n*3n(n-1)/2
    10. No. of asymmetric relations = 3n(n-1)/2
    11. No. of irreflexive relations = 2n(n-1)
    12. A relation is a partial order if
        1) Reflexive
        2) Antisymmetric
        3) Transitive
    13. Meet Semi Lattice :
        For all a, b belongs to L a∧b exists 
    14. Join Semi Lattice
        For all a, b belongs to L a∨b exists 
    15. A poset is called Lattice if it is both meet and join semi-lattice
    16. Complemented Lattice : Every element has complement
    17. Distributive Lattice : Every Element has zero or 1 complement .
    18. Boolean Lattice: It should be both complemented and distributive. Every element has exactly one complement.
    19. A relation is an equivalence if
        1) Reflexive
        2) symmetric
        3) Transitive

    Graph Theory

    1. No. of edges in a complete graph = n(n-1)/2
    2. Bipartite Graph : There is no edges between any two vertices of same partition . In complete bipartite graph no. of edges =m*n
    3. Sum of degree of all vertices is equal to twice the number of edges.
    4. Maximum no. of connected components in graph with n vertices = n
    5. Minimum number of connected components =
    0 (null graph)
    1 (not null graph) 
    6. Minimum no. of edges to have connected graph with n vertices = n-1
    7. To guarantee that a graph with n vertices is connected, minimum no. of edges required = {(n-1)*(n-2)/2 } + 1
    8. A graph is euler graph if it there exists atmost 2 vertices of odd – degree
    9. Tree
        -> Has exactly one path btw any two vertices
        -> not contain cycle
        -> connected
        -> no. of edges = n -1
    10. For complete graph the no . of spanning tree possible = nn-2
      11. For simple connected planar graph
    • A graph is planar if and only if it does not contain a subdivision of K5 and K3, 3 as a subgraph.
    • Let G be a connected planar graph, and let n, m and f denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of G. Then n – m + f = 2.
    • Let G be a connected planar simple graph with n vertices and m edges, and no triangles. Then m ≤ 2n – 4.
    • Let G be a connected planar simple graph with n vertices, where n ? 3 and m edges. Then m ≤ 3n – 6.
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