Flashcards: Real Analysis and Elementary Functions — 83 cards

What is a mathematical assertion?

A statement that is unambiguously either true or false.

When is the conjunction A and B true?

Only when both A and B are true.

When is the inclusive disjunction A or B true?

When at least one of A and B is true.

What does the implication A ⇒ B require?

B to be true whenever A is true.

What is the contrapositive of A ⇒ B?

¬B ⇒ ¬A.

What does the existential quantifier ∃x mean?

At least one x satisfies the assertion.

What does negating ∃x,A(x) yield?

It is equivalent to ∀x,¬A(x).

What does a proof by contradiction assume?

It assumes ¬A and derives incompatible conclusions.

When is set A included in set B?

When every element of A is also an element of B.

What does the intersection A ∩ B contain?

Elements belonging to both A and B.

What does the union A ∪ B contain?

Elements belonging to at least one of A and B.

What is the Cartesian product A × B?

The set of ordered pairs (a,b) with a in A and b in B.

What defines a function f from E to F?

Each element x of E is related to at most one element f(x) of F.

How is the composition g∘f defined for functions f and g?

By (g∘f)(x) = g(f(x)) on the domain where f and g are defined.

What characterizes an injective application?

It gives distinct images to distinct inputs.

When is an application f:E→F bijective?

When every equation f(x)=y has exactly one solution in E.

What does Euclidean division guarantee for natural numbers a and b with b≠0?

There is a unique pair of integers q and r such that a = bq + r and r < b.

What are q and r in Euclidean division of a by b?

They are the quotient and remainder respectively.

What is unique about the prime factorization of every natural number n≥2?

It is a product of primes p₁ < ⋯ < pₖ raised to nonzero integer exponents αᵢ.

What two steps does a proof by induction require to establish P(n) for all natural numbers?

Initialization P(0) and hereditary implication P(n) ⇒ P(n+1).

What is the binomial formula for (a+b)^n?

$(a+b)^n=\sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

How is the number of combinations of p elements from n calculated?

$\binom{n}{p} = \frac{n!}{p!(n-p)!}$ for 0 ≤ p ≤ n.

How is the absolute value of a real number x defined?

$|x|=\max(-x,x)$, the nonnegative number equal to x if x≥0, else −x.

What defines the order relation $x\le y$ on $\mathbb{R}$?

$x\le y$ if and only if $y - x \in [0,+\infty[$.

Which properties make $\le$ a total order on $\mathbb{R}$?

It is reflexive, transitive, antisymmetric, and any two real numbers are comparable.

What effect does adding or subtracting the same real number have on an inequality?

It preserves the inequality.

How does multiplication by a nonnegative number affect an inequality's direction?

It preserves the inequality's direction.

How does multiplication by a nonpositive number affect an inequality's direction?

It reverses the inequality's direction.

What inequalities are equivalent to $|x| \le a$ for $a \ge 0$?

$-a \le x \le a$ is equivalent to $|x| \le a$.

What inequalities are equivalent to $|x| > a$ for $a \ge 0$?

$x < -a$ or $x > a$ is equivalent to $|x| > a$.

What distinguishes a maximum or minimum from a supremum or infimum?

A maximum or minimum belongs to the set; a supremum or infimum may not.

How is a real polynomial represented in terms of coefficients?

By an infinite real coefficient sequence that is zero from some rank onward.

What inequality relates degrees of nonzero polynomials A, B, and A+B?

$\deg(A+B)\le\max(\deg A,\deg B)$.

What is the formula for the degree of the product of two nonzero polynomials A and B?

$\deg(AB)=\deg A + \deg B$.

What does Euclidean division of polynomials A by nonzero B guarantee?

A unique pair $Q,R$ with $A=BQ+R$ and $\deg(R)<\deg(B)$.

How does polynomial Euclidean division proceed?

By repeatedly canceling the dominant term of the dividend using a multiple of the divisor until the remainder's degree is smaller.

When does a polynomial B divide a polynomial A?

If and only if the remainder of the Euclidean division of A by B is zero.

What is the condition for a real number a to be a root of polynomial P?

If and only if $X - a$ divides $P$.

What is the maximum number of distinct real roots of a nonzero polynomial of degree n?

At most $n$ distinct real roots.

How is the natural logarithm defined on positive real numbers?

By the integral $\ln(x)=\int_1^x \frac{1}{t} \, dt$.

What formula relates $\ln(ab)$ to $\ln a$ and $\ln b$ for positive $a,b$?

$\ln(ab) = \ln a + \ln b$.

What unique positive number $e$ satisfies $\ln e = 1$?

The number $e$ is the unique positive number with $\ln e = 1$.

What is the exponential function in relation to the natural logarithm?

It is the inverse function of the natural logarithm.

How is the real power $a^\alpha$ defined for $a>0$ and real $\alpha$?

By $a^\alpha = \exp(\alpha \ln a)$.

What is the derivative of the power function $x^\alpha$ for positive $x$?

It is $\frac{d}{dx} x^\alpha = \alpha x^{\alpha - 1}$.

What are $\cos x$ and $\sin x$ on the unit circle?

They are the abscissa and ordinate of the point for angle $x$.

What fundamental identity do sine and cosine satisfy?

$\cos^2 x + \sin^2 x = 1$.

What is the period of sine and cosine functions?

Their period is $2\pi$.

What are the derivatives of $\sin x$ and $\cos x$?

$\sin'(x) = \cos x$ and $\cos'(x) = -\sin x$.

How is the tangent function defined?

$\tan x = \frac{\sin x}{\cos x}$ on its domain.

What is the period of the tangent function?

Its period is $\pi$.

What is the derivative formula for $\tan x$?

$\tan'(x) = 1 + \tan^2 x = \frac{1}{\cos^2 x}$.

What are the addition formulas for sine and cosine?

$\cos(a+b) = \cos a \cos b - \sin a \sin b$ and $\sin(a+b) = \sin a \cos b + \cos a \sin b$.

What is the algebraic form of a complex number z?

z = x + iy with x,y real numbers.

What is the formula for adding two complex numbers z and z'?

z+z'=(x+x')+i(y+y').

What is the formula for multiplying two complex numbers z and z'?

zz'=(xx'-yy')+i(xy'+x'y).

How is a complex number z represented in the complex plane?

By the point M(x,y) or vector x⃗i + y⃗j in an orthonormal system.

What is the affix of the vector AB if A and B have affixes zA and zB?

It is zB - zA.

What is the conjugate of z = x + iy?

The complex number z̄ = x − iy.

How is the modulus of z = x + iy defined?

It is the nonnegative real number $|z|=\sqrt{x^2+y^2}$.

What equation relates a complex number z and its conjugate?

They satisfy $z\bar z=|z|^2=x^2+y^2$.

What is the formula for the inverse of a nonzero complex number z?

Its inverse is $\frac1z=\frac{\bar z}{|z|^2}=\frac{x-iy}{x^2+y^2}$.

What defines an argument of a nonzero complex number z?

Any real θ with $z=|z|(\cos\theta+i\sin\theta)$ is an argument of z.

How are all arguments of a nonzero complex number related?

They differ by $2k\pi$ for integers k.

What is the trigonometric form of any nonzero complex number?

It is $z=re^{i\theta}$ where $r=|z|$ and $\theta$ is an argument modulo $2\pi$.

How does the argument of a product of two nonzero complex numbers relate to their arguments?

It equals the sum of their arguments modulo $2\pi$.

How does the argument of a quotient of two nonzero complex numbers relate to their arguments?

It equals the difference of their arguments modulo $2\pi$.

What is the formula for the n-th roots of $z = re^{i\theta}$ with $r>0$?

They are $Z_k=\sqrt[n]{r}e^{i(\theta+2k\pi)/n}$ for $k=0,\ldots,n-1$.

How many distinct n-th roots does a nonzero complex number have?

Exactly n distinct n-th roots.

What condition defines a finite limit ℓ of f at a?

For every ε > 0, there exists α > 0 such that |x − a| ≤ α implies |f(x) − ℓ| ≤ ε.

When does a two-sided limit at a exist?

When the left-hand and right-hand limits exist and are equal at a.

What additional condition relates the two-sided limit and f(a) when f is defined at a?

The two-sided limit equals f(a) when f is defined at a.

What is the definition of continuity of f at a?

f is continuous at a if it is defined at a and $\lim_{x\to a}f(x)=f(a).$

What does the intermediate value property state for a continuous function on an interval?

If f is continuous and takes opposite signs, there exists x with f(x) = 0 in the interval.

What is the image of a segment [a,b] under a continuous function f?

f([a,b]) is a segment [m,M].

What does continuity on [a,b] imply about f's bounds?

f is bounded and attains its minimum m and maximum M on [a,b].

What does the squeeze theorem state about f(x) if h(x) ≤ f(x) ≤ g(x) near +∞?

f(x) tends to the same limit ℓ as h(x) and g(x).

Which function operations preserve continuity at a point?

Sums, products, scalar multiples, and reciprocals (if denominator nonzero) preserve continuity.

What condition ensures the composition g∘f is continuous at a?

f continuous at a, f(I)⊆J, and g continuous at f(a).

What determines the limit at ±∞ of a polynomial of degree n?

Its leading term a_nx^n determines the limit.

What is the limit at ±∞ of a reduced rational function P(x)/Q(x) when numerator degree n < denominator degree p?

The limit is 0.

What is the limit at ±∞ of P(x)/Q(x) when numerator degree n > denominator degree p?

The limit is infinite in the leading-term direction.

What is the limit at ±∞ of P(x)/Q(x) when numerator degree n equals denominator degree p?

The limit is the ratio of leading coefficients a_n/b_p.