1. Introduction
Discrete mathematics studies objects that are separate and countable — sets, statements, integers, graphs — rather than smooth continuous quantities. Its three foundational pillars are set theory (the language for collections of objects), logic (the rules for reasoning correctly), and proof (the method for establishing that a claim is true beyond doubt).
For BCS, Bank and IT-officer exams this topic is a goldmine: questions on Venn diagrams, the inclusion–exclusion principle, truth tables, tautologies, De Morgan's laws, quantifiers and simple proofs appear almost every year. The reasoning is exact, the answers are unambiguous, and once you master the small set of rules here you can solve the questions quickly and with full confidence. This lesson builds every idea from first principles so that you not only memorise formulas but truly understand why they work.
2. Learning objectives
After studying this lesson you will be able to:
- Define a set precisely and use set-builder and roster notation correctly.
- Classify sets (finite, infinite, empty, singleton, equal, equivalent, subset, power set, universal).
- Perform all set operations (union, intersection, difference, complement, symmetric difference, Cartesian product) and read/draw Venn diagrams.
- Apply set identities and the inclusion–exclusion principle to counting problems.
- Build truth tables and identify tautologies, contradictions and contingencies.
- Recognise logical equivalences (De Morgan, distributive, contrapositive) and simplify statements.
- Use quantifiers (∀, ∃) and correctly negate quantified statements.
- Test the validity of arguments using rules of inference.
- Write and recognise the four main proof methods: direct, contrapositive, contradiction and mathematical induction, and disprove claims with counterexamples.