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  • Particular Solution

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    (a) Homogeneous Linear Difference Equations and Particular Solution: We can find the particular solution of the difference equation when the equation is of homogeneous linear type by putting the values of the initial conditions in the homogeneous solutions. Example1: Solve the difference equation 2ar-5ar-1+2ar-2=0 and find particular solutions such that a0=0 and a1=1. Solution: The characteristics equation…

  • Linear Recurrence Relations with Constant Coefficients

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    A Recurrence Relations is called linear if its degree is one. The general form of linear recurrence relation with constant coefficient is           C0 yn+r+C1 yn+r-1+C2 yn+r-2+⋯+Cr yn=R (n) Where C0,C1,C2……Cn are constant and R (n) is same function of independent variable n. A solution of a recurrence relation in any function which satisfies the given…

  • Recurrence Relations

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    A recurrence relation is a functional relation between the independent variable x, dependent variable f(x) and the differences of various order of f (x). A recurrence relation is also called a difference equation, and we will use these two terms interchangeably. Example1: The equation f (x + 3h) + 3f (x + 2h) + 6f (x…

  • The Pigeonhole Principle

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    If n pigeonholes are occupied by n+1 or more pigeons, then at least one pigeonhole is occupied by greater than one pigeon. Generalized pigeonhole principle is: – If n pigeonholes are occupied by kn+1 or more pigeons, where k is a positive integer, then at least one pigeonhole is occupied by k+1 or more pigeons.…

  • Permutation and Combinations

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    Permutation: Any arrangement of a set of n objects in a given order is called Permutation of Object. Any arrangement of any r ≤ n of these objects in a given order is called an r-permutation or a permutation of n object taken r at a time. It is denoted by P (n, r)   …

  • Basic Counting Principles

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    Sum Rule Principle: Assume some event E can occur in m ways and a second event F can occur in n ways, and suppose both events cannot occur simultaneously. Then E or F can occur in m + n ways. In general, if there are n events and no two events occurs in same time then…

  • Normal Forms

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    The problem of finding whether a given statement is tautology or contradiction or satisfiable in a finite number of steps is called the Decision Problem. For Decision Problem, construction of truth table may not be practical always. We consider an alternate procedure known as the reduction to normal forms. There are two such forms: Disjunctive…

  • Predicate Logic

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    Predicate Logic deals with predicates, which are propositions, consist of variables. Predicate Logic – Definition A predicate is an expression of one or more variables determined on some specific domain. A predicate with variables can be made a proposition by either authorizing a value to the variable or by quantifying the variable.The following are some…

  • Tautologies and Contradiction

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    Tautologies A proposition P is a tautology if it is true under all circumstances. It means it contains the only T in the final column of its truth table. Example: Prove that the statement (p⟶q) ↔(∼q⟶∼p) is a tautology. Solution: Make the truth table of the above statement: p q p→q ~q ~p ~q⟶∼p (p→q)⟷( ~q⟶~p) T…

  • Conditional and BiConditional Statements

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    Conditional Statement Let p and q are two statements then “if p then q” is a compound statement, denoted by p→ q and referred as a conditional statement, or implication. The implication p→ q is false only when p is true, and q is false; otherwise, it is always true. In this implication, p is…