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The function composition $g\circ f:A\to C$ is defined by $(g\circ f)(a)=g(f(a))$. Suppose $f(a)=f(a^{\prime })$. Applying $g$ gives $(g\circ f)(a)=(g\circ f)(a^{\prime })$. Since the composite is an injective function, $a=a^{\prime }$. Thus $f$ is an injective function.

Place $N$ objects into $m\ge 1$ boxes, assigning each object to exactly one box. Here $N$ is a nonnegative integer and $m$ is a positive integer. The pigeonhole principle says that if $N>m$, some box contains at least two objects. Equivalently, a function from an $N$-element finite set to an $m$-element finite set cannot be an injective function when $N>m$. The boxes may stand for categories rather than physical containers. ^[ref]

For thirteen objects and twelve boxes, a box containing at least two is unavoidable. Thinking of thirteen people and their birth months gives the same conclusion: at least two share a birth month. Twelve people could occupy the twelve months one apiece, so thirteen is the smallest number that forces a shared month. The generalized pigeonhole principle quantifies how many objects must share a box when there are still more objects.

The useful surprise is that the pigeonhole principle proves existence without needing to locate the objects involved. Choosing a good assignment rule is often the creative step: equal labels can mean equal remainders, nearby points, or repeated states of an algorithm. Once the rule is chosen, a short count can force a relationship among a much larger collection of objects. The pigeonhole principle itself does not name the crowded box. When the objects and their assignments are explicitly available, a simple algorithm finds one: scan the objects, keep the first object seen in each box, and stop when a later object enters an occupied box. At most $m+1$ scanned objects are needed when $N>m$.

A geometric example. Any five points in a square of side one contain two whose Euclidean distance is at most $1/\sqrt{2}$. Give each point Cartesian coordinates $(x,y)\in [0,1{]}^2$. Divide both ranges into the intervals $[0,1/2)$ and $[1/2,1]$, making four cells. Thus $x<1/2$ belongs to the left half and $x\ge 1/2$ to the right half; $y<1/2$ belongs to the lower half and $y\ge 1/2$ to the upper half. This convention assigns every boundary point to exactly one cell.

The pigeonhole principle puts two of the five points in the same cell. Their differences in Cartesian coordinates have magnitude at most $1/2$, so the Pythagorean theorem gives their Euclidean distance $d$ the boundFor the plotted points $(0.1,0.1)$, $(0.4,0.4)$, $(0.8,0.2)$, $(0.2,0.8)$ and $(0.8,0.8)$, the first two occupy the lower-left cell and have Euclidean distance $\sqrt{0.3^2+0.3^2}=\sqrt{0.18}\approx 0.424264$. These Cartesian coordinates exhibit the forced pair; the argument applies to every placement of five points.

An arithmetic example. Assign eight integers to their seven residue classes modulo the integer modulus $7$. The pigeonhole principle gives two with the same residue class, so $7$ is an integer divisor of their difference. In $\{2,5,9,12,16,20,24,27\}$, for example, $2$ and $9$ satisfy the integer congruence $9\equiv 2(\mathrm{mod}7)$, and $9-2=7$. Likewise, assigning fractional parts to short intervals gives the Dirichlet approximation theorem; its existing article explains the approximation bound.

A natural transformation between functors to the Category of sets is a monomorphism in the functor category exactly when every component is an injective function. Componentwise cancellation proves sufficiency. For necessity, the Yoneda lemma converts two elements with equal component images into two transformations from a covariant representable functor; monicity cancels these transformations, forcing the elements to coincide. The argument remains valid for a locally small category in an ambient universe where the functor category is formed.

For finite sets, the number of ordered choices of one element from each is the product of the individual numbers of choices. More generally, if stage $j$ has exactly $n_j$ possible choices after every permitted history of earlier stages, the total number of full choice sequences is ${\prod }_j{n}_j$, by mathematical induction on the stages. This counts $b^a$ functions between sets of sizes $a,b$, and counts the injective functions by the falling factorial $b(b-1)\cdots (b-a+1)$ when $a\le b$.

With five people, choose a captain in five ways and a different deputy in four ways: $5\cdot 4=20$ ordered assignments. The second choice depends on the first because the captain is excluded, but its number of permitted options is always four. The rule of product therefore applies without any assumption about independent events. To count two-person teams instead, each subset has two possible role assignments, so division by $2!=2$ gives ten teams. This division works because every team was counted equally often.

A simple curve is an injective curve, apart from the permitted coincidence of its two endpoints when it is closed.

For an injective function that does not increase distances, one stretch-factor convention measures the largest compression that must be undone:If the sharp Lipschitz constant of $f$ is one, this equals bi-Lipschitz distortion. Fixing the scale matters: uniform compression $f(x)=x/2$ has stretch factor two in this convention, but bi-Lipschitz distortion one. In a geometric spanner, the corresponding stretch factor is the largest graph-route distance divided by direct Euclidean distance; the graph-to-Euclidean identification does not increase distances. ^[ref].

A thermometer measures temperature through a physical response whose relation to temperature is known. Examples include the length change from thermal expansion, a change in electrical resistance, and the pressure of a gas thermometer. If the response function $X=f(T)$ is an injective function over the working range, its measured value determines $T=f^{-1}(X)$ through the inverse function. Thermometer calibration establishes that relation for the particular instrument and measurement conditions. Two matching readings are meaningful only after calibration to a common scale and with the probes sampling comparable equilibrium states. ^[ref]

For a contact thermometer, the sensing part must reach thermal equilibrium with the region being measured. That region may differ from the rest of the object: an immersed probe near a heater can read a local temperature while the surrounding water remains cooler. A stable reading can also be a steady state set by competing heat transfer paths to the sample and the room. Good contact, adequate immersion, mixing where appropriate, and checks for spatial differences help establish which temperature the reading represents. These are measurement controls inferred from the thermal equilibrium requirement, not a guarantee that every flat trace is equilibrium.

A thermometer also changes the sample it touches. Consider an original toy calculation with an initially uniform sample at $T_o=350\mathrm{K}$, a probe at $T_p=300\mathrm{K}$, and constant whole-body heat capacities $C_o=100\mathrm{J}{\mathrm{K}}^{-1}$ and $C_p=1\mathrm{J}{\mathrm{K}}^{-1}$. Suppose together they form an isolated system, with no work or phase transition and negligible contact energy. The first law of thermodynamics and thermal equilibrium giveThe final thermometer reading underestimates the sample's original temperature by about $0.495\mathrm{K}$, although it correctly measures their new common temperature. The disturbance becomes small when the participating probe heat capacity is small compared with the sample's. A large controlled heat bath is a different limit: it supplies or removes heat while approximately maintaining its own temperature.

Waiting time is a separate issue from disturbance. Under a toy Newton's law of cooling model for a uniform probe in a maintained heat bath, a constant contact conductance $G$ givesThe time constant $\tau$ controls this exponential decay. For stipulated values $T_b=350\mathrm{K}$, $T_p(0)=300\mathrm{K}$ and $\tau =5\mathrm{s}$, the probe reads about $331.61\mathrm{K}$ after one time constant and $347.51\mathrm{K}$ after three. These are calculated values, not an instrument specification or measured data. Real probes may have multiple relaxation times and additional heat transfer paths; choose an equilibration criterion from the required accuracy, rather than assuming that a fixed waiting time works universally.

For nonzero coprime integers $p,q$, the torus knot $T_{p,q}$ lies on a standard unknotted torus in the three-sphere and winds $p$ times meridionally and $q$ times longitudinally. Here a meridian of a solid torus goes around the small cross-section of the doughnut, while the longitudinal direction goes around its central hole. The integers count signed turns, so changing one sign gives the mirror of a link of the torus knot. In this article $p$ is always the meridional count and $q$ the longitudinal count.

An explicit parametric curve in three-dimensional Euclidean space iswhere $R>r>0$. Its cylindrical radius $\rho =\sqrt{x^2+y^2}$ satisfies $(\rho -R{)}^2+z^2=r^2$, the equation of the supporting torus. Its meridional angle is $pt$ and its longitudinal angle is $qt$: this checks the convention directly rather than relying on the picture. Its squared speed isIf two parameter values give the same point, their difference $\delta$ satisfies $p\delta ,q\delta \in 2\pi \mathbb{Z}$. Bézout's identity for the coprime integers $p,q$ then gives $\delta \in 2\pi \mathbb{Z}$. Thus the loop closes and is an injective function on the parameter circle, with nonzero derivative; it is a smooth embedding.

For example, $R=2$, $r=0.7$, $p=2$, $q=3$ produces a trefoil knot. During one circuit the meridional angle changes by $4\pi$ and the longitudinal angle by $6\pi$. The supporting torus is a guide for construction, not a constraint on later ambient isotopies: the knot may move off it. This family makes arithmetic visible as geometry: the coprime integers determine whether a single traversal visits the whole loop without repetition.

The coprime integers hypothesis matters. If the greatest common divisor is $d=\gcd (|p|,|q|)>1$, the same parametric curve retraces a single component $d$ times; it does not automatically draw $d$ different components. The corresponding torus link is instead the full set of points on the torus whose angles satisfy $q\varphi -p\theta \in 2\pi \mathbb{Z}$, with meridional angle $\varphi$ and longitudinal angle $\theta$. It has $d$ disjoint components, each with winding counts $(p/d,q/d)$. For instance, the torus link $T_{2,2}$ consists of two parallel $(1,1)$ loops and is a Hopf link, whereas substituting $(2,2)$ into ${\gamma }_{p,q}$ simply traces one of those loops twice.

Fix $n\ge 1$. There is no injective function from all length-$n$ binary strings to binary strings of length strictly less than $n$. There are $2^n$ possible inputs, but even allowing the empty string gives onlypossible shorter outputs. The finite geometric series and pigeonhole principle leave at least two inputs sharing an output, which prevents exact decoding. For $n=3$, eight inputs compete for the empty string and the six strings 0, 1, 00, 01, 10, 11.

This counting argument includes any file header or other information sent to the decoder. An algorithm chosen separately for each input must also be identified to the receiver; its description is not free. A useful compressor saves space on some inputs while leaving others unchanged or making them longer. Lossy compression avoids the exact-recovery requirement by deliberately allowing several inputs to share an output.

Put a contact thermometer $C$ in thermal contact with $A$, wait for thermal equilibrium, and record response $X_A$. Repeat with $B$ under the same operating constraints. If $X_A=X_B$, and the calibrated response is an injective function of temperature on this range, $C$ occupies the same thermal reference state in both comparisons. The zeroth law of thermodynamics then predicts $A\sim B$. Equality of raw numbers from different uncalibrated thermometers supplies no such premise.^[ref]

Use a probe whose participating heat capacity is small relative to each sample, or use maintained heat baths. Otherwise the matching readings describe the disturbed final states. Also hold the probe's immersion and mechanical conditions fixed: its response can depend on more than temperature. Matching readouts only tests agreement within resolution and repeatability, rather than proving exact equality of temperatures.

Agreement within a tolerance is not itself an equivalence relation. With a $0.10{}^{\circ }\mathrm{C}$ tolerance, readings $25.00$ and $25.16{}^{\circ }\mathrm{C}$ each agree with $25.08{}^{\circ }\mathrm{C}$, yet disagree with each other. This is a limitation of the measurement rule, not a failure of the exact transitivity asserted by the zeroth law of thermodynamics.