An ordered pair (a,b)(a,b)(a,b) is an object determined by two entries aaa and bbb such that (a,b)=(c,d)⟺(a=c) ∧ (b=d).(a,b)=(c,d)\quad\Longleftrightarrow\quad (a=c)\ \land\ (b=d).(a,b)=(c,d)⟺(a=c) ∧ (b=d).Ordered pairs are the building blocks of Cartesian products, graphs of functions, and relations. The defining property above is what distinguishes an ordered pair from a 2-element set, where order is irrelevant. Examples: (1,2)≠(2,1)(1,2)\neq(2,1)(1,2)=(2,1). In R2\mathbb{R}^2R2, a point is an ordered pair (x,y)(x,y)(x,y) with x,y∈Rx,y\in\mathbb{R}x,y∈R. The graph of f(x)=x2f(x)=x^2f(x)=x2 is {(x,x2):x∈R}⊆R×R\{(x,x^2):x\in\mathbb{R}\}\subseteq\mathbb{R}\times\mathbb{R}{(x,x2):x∈R}⊆R×R.

In Rk\mathbb{R}^kRk with the standard inner product, two vectors x,y∈Rkx,y\in\mathbb{R}^kx,y∈Rk are orthogonal, written x⊥yx\perp yx⊥y, if ⟨x,y⟩=0.\langle x,y\rangle = 0.⟨x,y⟩=0.Orthogonality generalizes the geometric notion of “perpendicular.” It is a key concept in decompositions (e.g., Pythagorean theorem, orthonormal bases) and in analysis via projections and least squares. Examples: In R2\mathbb{R}^2R2, (1,0)⊥(0,1)(1,0)\perp(0,1)(1,0)⊥(0,1). In R3\mathbb{R}^3R3, (1,1,0)⊥(1,−1,0)(1,1,0)\perp(1,-1,0)(1,1,0)⊥(1,−1,0) since 1⋅1+1⋅(−1)+0⋅0=01\cdot 1 + 1\cdot(-1)+0\cdot 0=01⋅1+1⋅(−1)+0⋅0=0. A nonzero vector xxx is never orthogonal to itself, since ⟨x,x⟩>0\langle x,x\rangle>0⟨x,x⟩>0.

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be bounded . For a subinterval I⊆[a,b]I\subseteq[a,b]I⊆[a,b], define the oscillation of fff on III by ω(f;I)=HAHAHUGOSHORTCODE733s2HBHBx∈If(x)−HAHAHUGOSHORTCODE733s3HBHBx∈If(x) ∈[0,∞). \omega(f;I)=sup _{x\in I} f(x)-inf _{x\in I} f(x)\ \in[0,\infty). ω(f;I)=HAHAHUGOSHORTCODE733s2HBHBx∈I​f(x)−HAHAHUGOSHORTCODE733s3HBHBx∈I​f(x) ∈[0,∞). ...

Let f:E→Rf:E\to\mathbb{R}f:E→R be a bounded function and let A⊆EA\subseteq EA⊆E be nonempty. The oscillation of fff on AAA is osc⁡(f;A):=HAHAHUGOSHORTCODE734s1HBHB{f(x):x∈A}−HAHAHUGOSHORTCODE734s2HBHB{f(x):x∈A}.\operatorname{osc}(f;A):=sup \{f(x):x\in A\}-inf \{f(x):x\in A\}.osc(f;A):=HAHAHUGOSHORTCODE734s1HBHB{f(x):x∈A}−HAHAHUGOSHORTCODE734s2HBHB{f(x):x∈A}.For Riemann integration , oscillation on subintervals controls the gap between upper and lower sums : on an interval III, the contribution to U(f,P)−L(f,P)U(f,P)-L(f,P)U(f,P)−L(f,P) is the oscillation on III times the interval length. ...

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open , let f:U→Rf:U\to\mathbb{R}f:U→R be a scalar-valued function, and let a=(a1,…,ak)∈Ua=(a_1,\dots,a_k)\in Ua=(a1​,…,ak​)∈U. The partial derivative of fff with respect to the jjjth variable at aaa is ∂f∂xj(a):=lim⁡h→0f(a1,…,aj+h,…,ak)−f(a1,…,aj,…,ak)h, \frac{\partial f}{\partial x_j}(a) :=\lim_{h\to 0}\frac{f(a_1,\dots,a_j+h,\dots,a_k)-f(a_1,\dots,a_j,\dots,a_k)}{h}, ∂xj​∂f​(a):=h→0lim​hf(a1​,…,aj​+h,…,ak​)−f(a1​,…,aj​,…,ak​)​, provided the limit exists. ...

A partial order on a set XXX is a relation ⪯ ⊆X×X\preceq\ \subseteq X\times X⪯ ⊆X×X such that for all x,y,z∈Xx,y,z\in Xx,y,z∈X: (Reflexive) x⪯xx\preceq xx⪯x. (Antisymmetric) If x⪯yx\preceq yx⪯y and y⪯xy\preceq xy⪯x, then x=yx=yx=y. (Transitive) If x⪯yx\preceq yx⪯y and y⪯zy\preceq zy⪯z, then x⪯zx\preceq zx⪯z. A set equipped with a partial order is a partially ordered set (poset). Partial orders capture “comparison” that may leave some pairs incomparable. ...

Given a sequence (an)(a_n)(an​) in R\mathbb{R}R or C\mathbb{C}C, the partial sums of the series ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞​an​ are the numbers sN:=∑n=1Nan(N∈N).s_N := \sum_{n=1}^N a_n\qquad (N\in\mathbb{N}).sN​:=n=1∑N​an​(N∈N).A series is defined to converge precisely when its partial sums converge as N→∞N\to\inftyN→∞. Many convergence tests are statements about the behavior of (sN)(s_N)(sN​). ...

A partition of a set XXX is a collection P\mathcal{P}P of subsets of XXX such that: every P∈PP\in\mathcal{P}P∈P is nonempty, if P,Q∈PP,Q\in\mathcal{P}P,Q∈P and P≠QP\neq QP=Q, then P∩Q=∅P\cap Q=\varnothingP∩Q=∅ (pairwise disjointness), ⋃P∈PP=X\bigcup_{P\in\mathcal{P}} P = X⋃P∈P​P=X (covers all of XXX). Partitions encode “grouping” of elements. Every equivalence relation induces a partition into equivalence classes, and conversely every partition defines an equivalence relation by declaring two elements equivalent iff they lie in the same part. ...

A partition of a closed interval [a,b]⊆R[a,b]\subseteq\mathbb{R}[a,b]⊆R is a finite set of points written in increasing order P: a=x0<x1<⋯<xn=b.P:\ a=x_0<x_1<\cdots<x_n=b.P: a=x0​<x1​<⋯<xn​=b. The associated subintervals are [xi−1,xi][x_{i-1},x_i][xi−1​,xi​] for i=1,…,ni=1,\dots,ni=1,…,n, and their lengths are Δxi:=xi−xi−1\Delta x_i:=x_i-x_{i-1}Δxi​:=xi​−xi−1​. ...

Let (X,d)(X,d)(X,d) be a metric space. A path in XXX is a continuous function γ:[a,b]→X\gamma:[a,b]\to Xγ:[a,b]→X for some real interval [a,b]⊂R[a,b]\subset\mathbb{R}[a,b]⊂R (often [0,1][0,1][0,1]). The endpoints of the path are γ(a)\gamma(a)γ(a) and γ(b)\gamma(b)γ(b). Paths model “continuous motion” inside a space. Path-connectedness is stronger than connectedness and is often easier to verify in geometric settings. Examples: In Rk\mathbb{R}^kRk, the map γ(t)=(1−t)x+ty\gamma(t)=(1-t)x+tyγ(t)=(1−t)x+ty for t∈[0,1]t\in[0,1]t∈[0,1] is a path from xxx to yyy (a line segment). On the unit circle S1⊂R2S^1\subset\mathbb{R}^2S1⊂R2, γ(t)=(cos⁡t,sin⁡t)\gamma(t)=(\cos t,\sin t)γ(t)=(cost,sint) for t∈[0,2π]t\in[0,2\pi]t∈[0,2π] is a path with γ(0)=γ(2π)\gamma(0)=\gamma(2\pi)γ(0)=γ(2π). In the discrete metric space XXX (with at least two points), any continuous map from a connected interval [a,b][a,b][a,b] must be constant, so there are no nontrivial paths connecting distinct points.