Term 2 content (not 2.5) Flashcards
What are the algebraic properties associated with the binary operation +
[commutativity]x+y=y+x
associativity+z=x+(y+z)
[identity] there exists an element 0 ∈ R such that 0+x = x = x + 0, for every x ∈ R,
[inverses] for every x ∈ R, there exists w ∈ R such that x+w = 0 = w + x,
What are the algebraic properties associated with the binary operation *
[commutativity]x·y=y·x
associativity·z=x·(y·z)
[identity] there exists an element 1 ∈ R with 1 ̸= 0, such that 1 · x = x = x · 1, for every x ∈ R,
[inverses] for every x ̸= 0 in R, there exists u ∈ R such that x · u = 1 = u · x,
[distributivity]x·(y+z)=x·y+x·z
what is the uniqueness of the additive inverse
if u and v are additive inverses then u = v
what are the order properties
[trichotomy] for every x ∈ R, exactly one of the following holds:
x ∈ P , x = 0, −x
[compatibility with +] x+y ∈ P, for every x,y
[compatibility with ·] x·y ∈
what are the order properties using <, >
[trichotomy]for all x,y ∈ R exactly one of x<y or x=y or
y < x is true.
[transitivity]for all x,y,z∈R, if x<y and y<z, then x<z.
[compatibility with +] for all x,y,z ∈ R, if x < y, then x+z < y+z.
for all x,y,z∈R if x<y and 0<z,then z · x < z · y.
What is the completeness axiom
LetS⊆R.Anelementx∈Risanupperboundof S if s ≤ x, for every s ∈ S. Any subset of R with an upper bound is said to be bounded above.
Likewise, x ∈ R is a lower bound of S if x ≤ s, for every s ∈ S. Any subset of R with a lower bound is said to be bounded below.
A set S that is bounded above and bounded below is said to be bounded.
Every x ∈ R is an upper bound for the empty set
Every x ∈ R is an upper bound for the empty set
Let S be a subset of R that is bounded above. A number x is said to be a supremum or least upper bound of S if
x is an upper bound of S
for any y if y is an upper bound of S then x <= y
Let S and T be subsets of R with suprema. If S ⊆ T, then sup(S) ≤ sup(T).
Let S and T be subsets of R with suprema. If S ⊆ T, then sup(S) ≤ sup(T).
what is the completeness axiom
For any non-empty subset S of
R, if S has an upper bound, then S has a supremum.
what is the Archimedean property
For any x ∈ R, there exists n ∈ N such that x < n.
Let ε > 0 be a real number. Then there exists k ∈ N for which 1/k < ε.
when is a set of real numbers bounded
Let S be a set of real numbers. Then S is bounded if and only if there exists a natural number M such that, for every s ∈ S, we have |s| < M.
Let S be a subset of R that is bounded below. A number x is said to be a infimum or greatest lower bound of S if
x is a lower bound of s and
for any y y is a lower bound of s then y<= x
Let S be a set of real numbers. Let −S denote the set {−s : s ∈ S} . Then
S has an upper bound if and only if -S has a lower bound.
if S is non-empty then S has a supremum with x = sup(S) if and only if -S has an infimum with -x = inf(S)
what is thew definition of convergence
The ∞
sequence (xn)n=1 converges to x if for any real number ε > 0, there exists a natural number N such that for any natural number n, if n > N, then
|xn − x| < ε.
Let (an)n=1 be a convergent sequence of complex num- ∞∞
bers. Suppose that (an)n=1 converges to s and that (an)n=1 converges to t, then s = t.
Let (an)n=1 be a convergent sequence of complex num- ∞∞
bers. Suppose that (an)n=1 converges to s and that (an)n=1 converges to t, then s = t.
Let (zn)n=1 be a sequence of complex numbers. The se- quence (zn)∞n=1 converges to 0 if and only if the sequence (|zn|)∞n=1 converges to 0.
Let (zn)n=1 be a sequence of complex numbers. The se- quence (zn)∞n=1 converges to 0 if and only if the sequence (|zn|)∞n=1 converges to 0.
what are the two forms of the triangle inequality
|u+v|<=|u|+|v|
||u|-|v||<=|u-v|
what is the proof for the reverse triangle inequality
|𝑥|+|𝑦−𝑥|≥|𝑥+𝑦−𝑥|=|𝑦|
|𝑦|+|𝑥−𝑦|≥|𝑦+𝑥−𝑦|=|𝑥|
|𝑦−𝑥|≥|𝑦|−|𝑥|
|𝑥−𝑦|≥|𝑥|−|𝑦|.
|𝑦−𝑥|=|𝑥−𝑦|, and if 𝑡≥𝑎 and 𝑡≥−𝑎 then 𝑡≥|𝑎|.
Combining these two facts together, we get the reverse triangle inequality:
|𝑥−𝑦|≥∣∣|𝑥|−|𝑦|∣∣.
Let (an)n=1 be a sequence of real numbers. Let S={an :n∈N},
the set of values of terms of the sequence. The sequence (an)n=1 ∞
is bounded above if S is bounded above. The sequence (an)n=1 ∞
is bounded below if S is bounded below. A sequence (bn)n=1 of complex numbers is bounded if {|bn| : n ∈ N} is bounded. Let (an)n=1 be a sequence of complex numbers. If (an)n=1 ∞
is convergent, then (an)n=1 is bounded.
Let (an)n=1 be a sequence of complex numbers. If (an)n=1 ∞
is convergent, then (an)n=1 is bounded.
Let (an)n=1 be a sequence of complex numbers. If (an)n=1 ∞
is convergent, then (an)n=1 is bounded.
What is the definition of monotone convergence
A sequence (an)n=1 of real numbers is increasing if an ≤ an+1, for every n. It is decreasing if an ≥ an+1, for every n. It is monotone if it is increasing or decreasing.
Let (an)n=1 be a sequence of real numbers. If (an)n=1 ∞
is bounded and monotone, then (an)n=1 is convergent.
Let (an)n=1 be a sequence of real numbers. If (an)n=1 ∞
is bounded and monotone, then (an)n=1 is convergent.
