TU BSc CSIT Entrance cheat sheet

B.Sc.CSIT Entrance - Core Subject Reference

• Set theory: |A∪B| = |A| + |B| − |A∩B|; for three sets subtract pairwise intersections and add back the triple intersection. A subset of an n-element set has 2^n subsets total, of which 2^n − 1 are proper non-empty.

• Logic: p→q ≡ ¬p∨q ≡ ¬q→¬p (contrapositive). Negation of ∀x P(x) is ∃x ¬P(x); negation of ∃x P(x) is ∀x ¬P(x). A tautology is true under every valuation.

• Permutations and combinations: P(n,r) = n!/(n−r)!; C(n,r) = n!/(r!(n−r)!). Circular arrangements of n distinct objects = (n−1)!. Arrangements of a multiset = n!/(n1!·n2!·…).

• Sequences: arithmetic term a+(n−1)d, sum n/2·(2a+(n−1)d); geometric term ar^(n−1), sum a(r^n−1)/(r−1) for r≠1. Infinite geometric sum converges only when |r|<1.

• Probability: P(A∪B) = P(A)+P(B)−P(A∩B); P(A|B) = P(A∩B)/P(B). Independent events satisfy P(A∩B)=P(A)P(B). Bayes' theorem reverses conditional direction using prior probabilities.

• Matrices: a 2×2 matrix has inverse only when ad−bc ≠ 0, with A⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]]. A system Ax=b has a unique solution iff det(A) ≠ 0. Matrix multiplication is not commutative.

• Calculus: d/dx(x^n)=nx^(n−1); d/dx(sin x)=cos x; d/dx(e^x)=e^x; d/dx(ln x)=1/x. Product rule (uv)'=u'v+uv'; quotient rule (u/v)'=(u'v−uv')/v². ∫x^n dx = x^(n+1)/(n+1)+C for n≠−1.

• Number systems: binary 1011₂ = 11₁₀; hexadecimal digits A–F equal 10–15. To convert decimal to base b, divide repeatedly and read remainders upward. Two's complement negates a fixed-width binary number by inverting bits and adding 1.

• Complex numbers: i² = −1; modulus of a+bi is √(a²+b²); conjugate of a+bi is a−bi and (a+bi)(a−bi)=a²+b². De Moivre: (cos θ + i sin θ)^n = cos nθ + i sin nθ.