Q: What is the base step in induction?
A: Verify the statement for the first integer (usually n=1).
Q: What makes strong induction “strong”?
A: You assume the statement is true for all numbers up to n, not just for n.
Q: Define factorial n!
A: n! = 1 · 2 · … · n with 0! = 1.
Q: What is (n choose k)?
A: n! / [k!(n−k)!].
Q: State the binomial theorem.
A: (a+b)^n = Σ (n choose k) a^k b^{n−k}.
Q: Why is √2 irrational?
A: Assuming √2 = p/q leads to p and q both even, contradicting that the fraction is simplified.
Q: What is i²?
A: −1.
Q: Definition of modulus |z| of a+bi?
A: sqrt(a² + b²).
Q: Conjugate of a+bi?
A: a − bi.
Q: Polar form of a complex number?
A: z = r(cosθ + i sinθ).
Q: State De Moivre’s theorem.
A: (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ).
Q: When is A ⇒ B false?
A: When A is true and B is false.
Q: Negation of “∀x, P(x)”
A: “∃x such that ¬P(x)”.
Q: What is a bijection?
A: A function that is both injective and surjective.
Q: How to test injectivity?
A: Show f(a)=f(b) ⇒ a=b.
Q: Three properties of an equivalence relation?
A: Reflexive, symmetric, transitive.
Q: Group axioms?
A: Associativity, identity element, inverses, closure.
Q: What is a homomorphism?
A: A map f satisfying f(xy)=f(x)f(y).
Q: Define supremum.
A: Least upper bound of a set.
Q: If a set has a maximum, what is the supremum?
A: The maximum itself.
Q: Definition of convergence of a sequence?
A: a_n → L if for every ε>0, ∃N such that |a_n–L|<ε for n≥N.
Q: What is a Cauchy sequence?
A: Terms get arbitrarily close to each other.
Q: When does a geometric series converge?
A: When |r| < 1.
Q: Ratio test result?
A: Converges if limit ratio < 1.
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Geometry - Euclid Axioms5
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Geometry - Lemmas27
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Geometry - Definitions1
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Probability - Distributions16
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Foundation Of Mathematics - Peano Axioms5
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Foundations 125
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Foundations Series14
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Linear Algebra - Vector Space Axioms10
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Vectors in 3 Dimensional Space9
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Examples and Properties of Vector Spaces8
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Subspaces2
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Linear Transformations2
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Linear Algebra: Isomorphisms, Kernels and Images3
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Linear Combinations4
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Linear Algebra - Linear Dependence and Independence1
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Linear Algebra - Bases, Co-ordinates and Dimensions.11
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Foundations - Convergent Series Tests12
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Unknown Part B Geometry8
