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Mathematics

Matrices
  • Idempotent Matrix : For a matrix A , if we get A2 = A , then it is an idempotent matrix.
  • Nilpotent Matrix : For a matrix A, if at some m we get Am = 0, then it is called nilpotent matrix, and smallest value of m at which this condition is satisfied is called order of matrix.
  • Involutory Matrix : For a matrix A , if we get A2 = I , then it is called involutory matrix.
  • Singular Matrix : For a matrix A, if det(A) is zero, then it is called singular matrix.
  • Non- Singular Matrix: For a matrix A , if det(A) is non-zero , then it is called non - singular matrix.
  • Symmetric matrix : For a matrix , if AT = A , then it is symmetric matrix.
  • Skew- Symmetric matrix : For a matrix , if AT = -A , then it is skew - symmetric matrix.
  • Sum of Symmetric and skew symmetric matrix is twice the matrix.

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Petersan Graph
  • Petersan graph is a graph with 10 vertices and 15 edges.
  • It is a non-planar graph.
  • It has got hamiltonian path but not hamiltonain cycle.
  • It has got chromatic number 3.

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@shivani 

pls explain point 3 .what are the condition for checking hamilton path

start from a vertex and you can trace each vertex exactly once without repeating vertices but you cant reach to the starting vertex without repeating try it.

habibkhan's picture

One more point that can be added is this graph contains K5 and its isomorphic image as well..

Mathematical Logic : Rules of Inference
  • Precedence of logical operators is as follows:
    ¬  > ∧  >  ∨   > →  >  ↔
    where, ¬  , ∧  ,  ∨  is left associative and → ,  ↔ is right associative.
    Example: P →Q →R can be wriiten as  (P →(Q →R))
  • Following are the rules of inference:
  1. Modus Ponens: 
    P
    P
    ________
    Q
    ________
  2. Modus Tollens:
    ¬Q
    P->Q
    ________
    ¬P
    ________
  3. Disjunctive Syllolgism:
    P∨Q
    ¬P
    ________
    Q
    ________
  4. Constructive Dilemma:
    (PQ ) ∧ (RS )
    (P∨R )
    ________
    Q∨S
    ________
  5. Destructive Dilemma:
    (PQ ) ∧ (RS )
    (¬Q∨¬S)
    ________
    ¬P∨¬R
    ________
  6. Disjunctive Syllolgism:
    P∨Q
    ¬P
    ________
    Q
    ________

Example:
If child studies , mom will not scould
Mom scoulded
Inference: Child did not study.
Ans. p =  child studies
q = Mom scould
p->~q
q
_____
~p          ///using Modus Tollens
_____
~p means child did not study.
Thus , given inference is true.

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Mathematics

Following are the topics to be covered in the mathematics section:

  • Linear Algebra
  • Calculus
  • Combinatory
  • Probability
  • Numerical Methods
  • Graph Theory
  • Set Theory & Algebra
  • Mathematical Logic

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