Engineering Mathematics and Discrete Mathematics
This domain tests the mathematical maturity behind computer science: linear algebra, calculus, probability, and numerical methods, plus discrete structures such as logic, sets, relations, functions, combinatorics, recurrence relations, algebraic structures, and graph theory. ISRO questions usually reward exact definitions and short computations rather than long proofs.
Core rules and concepts
- A relation on A is a subset of A×A; it is reflexive, symmetric, transitive, or antisymmetric by checking every ordered pair. A partial order is reflexive, antisymmetric, transitive; a lattice adds least upper and greatest lower bounds for every pair.
- A function is a relation with exactly one image per domain element. Injective, surjective, and bijective are independent of size only for finite sets.
- A group needs closure, associativity, identity, and inverses; a ring adds a second operation, and a field requires multiplicative inverses for nonzero elements.
- In graph theory, the sum of degrees equals twice the number of edges; a tree on n vertices has n−1 edges and is connected and acyclic.
- For combinatorics, use inclusion–exclusion for “at least one” constraints and solve linear recurrences via characteristic equations.
Worked example
Count onto functions from a 4-element set to a 3-element set. Total functions are 3^4=81. Subtract those missing at least one codomain element: choose the missing element in C(3,1)=3 ways, leaving 2^4=16 functions, so 48. Add back functions missing two elements: C(3,2)=3 ways, each leaving 1^4=1, so 3. Inclusion–exclusion gives 81−48+3=36 onto functions.
Traps that cost marks
Confusing a relation with a function; forgetting diagonal pairs when testing symmetry; assuming every relation is transitive; treating necessary conditions as sufficient; and misreading “at least one” as “exactly one.” In linear algebra, check whether eigenvalues are repeated before concluding diagonalizability.