MATHS Examples

One idea in several languages

A falling object has a height function $s(t)$. Set theory says $t$ must belong to a Domain, commonly an Interval such as $[0,T]$. Function language says $s:[0,T]\to \mathbb{R}$. Calculus says $v(t)=s^{\prime }(t)$ is the instantaneous Velocity. Physics then attaches units, e.g. metres and seconds, so $v$ is measured in m s${}^{-1}$.

Domain restriction makes an inverse possible

The rule $f(x)=x^2$ is not one-to-one on $\mathbb{R}$ because $f(2)=f(-2)=4$. Restrict the domain to $[0,\infty )$ and the same rule becomes injective and surjective onto $[0,\infty )$. Then $f^{-1}(y)=\sqrt{y}$ is a genuine inverse function.

Sets clarify probability language

For a die roll, the sample space is $\Omega =\{1,2,3,4,5,6\}$. The event “even” is the subset $E=\{2,4,6\}$. The probability function maps events to real numbers: $P(E)=3/6=1/2$. This is the same habit used in Stats index, just with larger and often continuous spaces.

Vector calculus bridge

A temperature field is a function from space to real numbers, $T:{\mathbb{R}}^3\to \mathbb{R}$. Its gradient $ablaT$ points in the direction of fastest increase. This same notation becomes physical in heat flow, fluids, and electromagnetism.