Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability


If a composite proposition p is always true for all possible assignment of truth values to its prime components, then it is called tautology.

The following are some examples of tautology:

  1. p \cup \left( { \sim p} \right)(Law of excluded middle)
  2.  \sim \left( {p \cap \sim p} \right)(Law of contradiction)
  3. p \leftrightarrow \sim \left( { \sim p} \right)(Law of double negation). (The above 3 tautologies are widely known laws of classical logic.)
  4. p \cap \left( {p \to q} \right) \to q
  5. \left[ {\left( {p \to q} \right) \cap \left( {p \to r} \right)} \right] \to \left[ {p \to q \cap r} \right]
  6. \left[ {\left( {p \to q} \right) \cap \left( {q \to r} \right)} \right] \to \left[ {p \to r} \right](principle of syllogism)
  7. \left[ { \sim q \to  \sim p} \right] \leftrightarrow \left[ {p \to q} \right](law of contrapositive)

If a condition p \to q is a tautology, then we say that p implies q i.e p \Rightarrow q.

Truth table

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