Combinatorics - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit II: Combinatorics
Counting problems arise throughout Mathematics and Computer Science. The two basic counting principles are 1. The product rule, 2. The sum rule
Combinatorics - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit II: Combinatorics
It is sometimes convenient to replace the induction hypothesis P (k) by the stronger assumption P (1), P (2), P (3), ... P (k) are true. The resulting principle known as the principle of strong mathematical induction.
Combinatorics - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit II: Combinatorics
The word induction means the method of inferring a general statement from the validity of particular cases. Mathematical induction is a technique by which one can prove mathematical statements involving positive integers.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
An exhaustive proof is a special type of proof by cases where each involves checking a single example.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
Direct proofs lead from the hypothesis of a theorem to the conclusion. In a direct proof, we assume that P is true and use axioms, definitions, and previous theorems, together with rules of inference, to show that must also be true.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
A formula S may be introduced in a derivation if S is tautologically implied by one or more of the preceeding formulae in the derivation.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
The main function of logic is to provide rules of inference, or principles of reasoning.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
Nested quantifiers are propositional functions where one or more quantifiers occurs within the scope of another quantifier.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
Let A and B be any two predicate formulas defined over a common universe denoted by the symbol E.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
Statements involving variables, such as "x > 4", "x = y+4", and "x + y = z", are often found in mathematical assertions and in computer programs.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
By constructing and comparing truth tables we can determine whether two statement formulas A and B are equivalent.
Logic and Proofs - Discrete Mathematics
Subject and UNIT: Discrete Mathematics: Unit I: Logic and Proofs
Compound propositions that have the same truth values in all possible cases are called logically equivalent.