a binary relation is an equivalence relation if

• \(R\) is reflexive: \(xRx\) is true for all \(x\)
• \(R\) is symmetric: \(x R y\) is true implies \(y R x\) is true
• \(R\) is transitive: for every \(x,y,z\), \(xRy\) and \(y R z\) implies \(x R z\)

A corollary of greatest common divisor and division.

Say you have some \(b|a\) such that:

\begin{equation} a = bq + r \end{equation}

Now, \(d|a,b \Leftrightarrow d|b,r\) (because \(d|b,r\) implies there’s some \(x, x’\) such that \(a = (dx)q+dx’\), and so \(a = d(xq + x’)\) and so \(d|a\); the logic goes the other way too).

This finally implies that \(\gcd (a,b)= \gcd (b,r)\) because any divisor that works for one works for both.

A type of cell.

Sample eukareotyic cell gene:

• TATA box promoter
• 5’ non-coding sequence
• Non-coding introns interlaced between exons, unique to eukareotyic cells. Bacteria (prokateotic cells don’t contain introns or have small them)
• 3’ non-coding sequence