Examples

Examples used in the Dictionary of Fine Distinctions. Each example is marked with the lowest level at which it is appropriate. Choosing a level shows the examples up to that level.

Elementary

• Appending $4$ to $(1, 2, 3)$ produces $(1, 2, 3, 4)$
  Terms: append; ; prepend; ;
• $A = \lbrace 1, 2 \rbrace$ and $B = \lbrace 3, 4 \rbrace$
  Terms: union; intersection; set difference; disjoint union (of disjoint sets);
• The location of Paris
  Terms: ; vector;
  Distinctions: Vector vs. point

High school

• $(0, 0) \in \reals^2$, for vector addition
  Terms: ; ; zero;
• In floating point arithmetic, $0.1 + 0.2$ evaluates to $0.30000000000000004$ due to rounding errors.
  Terms: ; ; bug;
• $0$, for the set $S = (0, 1]$
  Terms: ; ; ;
• $0$, for $x \mapsto x^2$
  Terms: ;
• $0 < 0.5 < 1$, for the interval $[0, 1]$
  Terms: ; ;
• The list of numbers $(1, 3, -2.4)$, with no other information
  Terms:
• $(2, 3) \in \reals^2$
  Terms: vector; ;
  Distinctions: Vector vs. point
• A velocity, in meters per second
  Terms: ; vector; covector;
  Distinctions: Vector vs. covector
• The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)
  Terms: array; matrix; linear map;
  Distinctions: Array vs. matrix vs. linear map
• The Euclidean norm $x \mapsto \sqrt{x_1^2 + x_2^2}$ on $\reals^2$

  The Euclidean norm, which gives the length of a vector in the plane.

  Terms: ; covector;
  Distinctions: Vector vs. covector
• The set of rational numbers, $\mathbb{Q}$
  Terms: ; ; ; ; ; ; ; ; ;
• The set of real numbers, $\reals$
  Terms: ; ; ; ; ; ; ; ;
• $\reals^3$
  Terms: ;
• Rotation of the plane by $90°$
  Terms: linear map; matrix; array;
  Distinctions: Array vs. matrix vs. linear map
• $A = \lbrace \text{apple} \rbrace$ and $B = \lbrace \text{banana} \rbrace$
  Terms: union; disjoint union (of disjoint sets); disjoint union (tagged);
  Distinctions: Disjoint union: union of disjoint sets vs. tagged union
• $A = \lbrace \text{apple}, \text{orange} \rbrace$ and $B = \lbrace \text{orange}, \text{banana} \rbrace$
  Terms: union; intersection; disjoint union (of disjoint sets); disjoint union (tagged);
  Distinctions: Disjoint union: union of disjoint sets vs. tagged union
• The row vector $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
  Terms: covector;
  Distinctions: Vector vs. covector
• The temperature gradient, in degrees per meter
  Terms: covector;
  Distinctions: Vector vs. covector

Undergraduate

• Differentiation, on polynomials of degree at most $2$
  Terms: linear map; matrix;
  Distinctions: Array vs. matrix vs. linear map
• $f(x, y) = x^2 + 3y$
  Terms: ; gradient; Jacobian; directional derivative; differential; Lie derivative;
  Distinctions: Types of multivariable derivatives

Graduate

• $[0, 1]$, for $\dot{x} = u$ with $\lvert u \rvert \leq 1$
  Terms: ; ;