GATE CS & IT Engineering Engineering Mathematics Syllabus

Engineering Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Engineering Mathematics in GATE CS & IT Engineering, not a summary of it.

1. 1

   Discrete Mathematics

   4 topics

   • Propositional and first order logic
   • Sets, relations, functions, partial orders and lattices
   • Monoids, Groups
   • Graphs
     • Connectivity
     • Matching
     • Coloring
2. 2

   Combinatorics

   3 topics

   • Counting
   • Recurrence relations
   • Generating functions
3. 3

   Linear Algebra

   5 topics

   • Matrices
   • Determinants
   • System of linear equations
   • Eigenvalues and eigenvectors
   • LU decomposition
4. 4

   Calculus

   5 topics

   • Limits
   • Continuity and differentiability
   • Maxima and minima
   • Mean value theorem
   • Integration
5. 5

   Probability and Statistics

   4 topics

   • Random variables
   • Uniform, normal, exponential, poisson and binomial distributions
   • Mean, median, mode and standard deviation
   • Conditional probability and Bayes theorem

Engineering Mathematics flashcards for GATE CS & IT Engineering

21 of 54 cards from the Engineering Mathematics deck — real questions with worked answers.

1. In propositional logic, when is the implication $p \to q$ false?

   Only when $p$ is true and $q$ is false. In all other cases ($p$ false, or $q$ true) the implication is true.

2. State the two De Morgan's laws for logical connectives.

   $\neg(p \land q) \equiv \neg p \lor \neg q$ and $\neg(p \lor q) \equiv \neg p \land \neg q$.

3. Write the contrapositive of $p \to q$ and state its relationship to the original.

   The contrapositive is $\neg q \to \neg p$. It is logically equivalent to $p \to q$.

4. Give the logical equivalence that rewrites an implication using only $\neg$ and $\lor$.

   $p \to q \equiv \neg p \lor q$.

5. In first-order logic, how do you negate $\forall x\, P(x)$ and $\exists x\, P(x)$?

   $\neg \forall x\, P(x) \equiv \exists x\, \neg P(x)$ and $\neg \exists x\, P(x) \equiv \forall x\, \neg P(x)$.

6. Define a tautology, a contradiction, and a contingency in propositional logic.

   A tautology is true under every truth assignment; a contradiction is false under every assignment; a contingency is true under at least one and false under at least one assignment.

7. What three properties must a relation satisfy to be an equivalence relation?

   Reflexivity ($a\,R\,a$), symmetry ($a\,R\,b \Rightarrow b\,R\,a$), and transitivity ($a\,R\,b \land b\,R\,c \Rightarrow a\,R\,c$).

8. What three properties define a partial order relation?

   Reflexivity, antisymmetry ($a\,R\,b \land b\,R\,a \Rightarrow a=b$), and transitivity.

9. How many distinct functions and how many injective functions exist from a set of size $m$ to a set of size $n$?

   Total functions: $n^{m}$. Injective (one-to-one) functions: $\frac{n!}{(n-m)!}$ for $m \leq n$ (else $0$).

10. Define a lattice in terms of a partially ordered set.

    A poset is a lattice if every pair of elements has both a least upper bound (join, $\lor$) and a greatest lower bound (meet, $\land$).

11. By the principle of inclusion-exclusion, what is $|A \cup B \cup C|$?

    $|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$.

12. How many distinct equivalence relations on a set correspond to its partitions, and what number counts them?

    Equivalence relations on a set are in one-to-one correspondence with its partitions; the number of partitions of an $n$-element set is the Bell number $B_n$.

13. What are the defining axioms of a group $(G,\ast)$?

    Closure, associativity, existence of an identity element $e$, and existence of an inverse for every element. (A commutative group additionally satisfies $a\ast b = b\ast a$.)

14. Distinguish a monoid from a group.

    A monoid is a set with an associative binary operation and an identity element. A group is a monoid in which every element additionally has an inverse.

15. State Lagrange's theorem for finite groups.

    The order of any subgroup $H$ of a finite group $G$ divides the order of $G$: $|H| \mid |G|$.

16. What is the order of an element $a$ in a group, and how does it relate to the identity?

    The order of $a$ is the smallest positive integer $n$ such that $a^{n}=e$. By Lagrange's theorem it divides $|G|$.

17. State the Handshaking Lemma for graphs.

    The sum of all vertex degrees equals twice the number of edges: $\sum_{v} \deg(v) = 2|E|$. Consequently the number of odd-degree vertices is even.

18. How many edges does a complete graph $K_n$ have, and what is the degree of each vertex?

    $K_n$ has $\binom{n}{2}=\frac{n(n-1)}{2}$ edges, and every vertex has degree $n-1$.

19. What is the relationship between vertices $V$, edges $E$, and faces $F$ in a connected planar graph (Euler's formula)?

    $V - E + F = 2$.

20. State the necessary and sufficient condition for a connected graph to have an Eulerian circuit.

    A connected graph has an Eulerian circuit if and only if every vertex has even degree.

21. When does a connected graph have an Eulerian path (but not necessarily a circuit)?

    When exactly zero or two vertices have odd degree. If two are odd, the path starts at one odd vertex and ends at the other.

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