# MAT385 Equivalence Classes Example

## Acknowledgment

This project is modified from

Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles
Brian Hopkins, editor
323 pp., paperbound, 2009. Series: MAA Notes #74
ISBN: 978-0-88385-184-5
http://www.maa.org/news/121808resources.html

It was provided to me for my use by Brian (who created it), after I saw him present the example at the 2009 Joint Mathematics Meetings in Washington, D.C. Thanks Brian!

## Codon Classes

Ribonucleic acid (RNA) is used by cells to produce proteins, among other tasks. RNA can be thought of as a string of nucleotide bases drawn from the set adenine, cytosine, guanine, and uracil. A codon is an ordered triple of bases that signals for a certain amino acid to be included in the sequence; for example, the codon AUG signals for methionine. There are ${\displaystyle 4^{3}=64}$ possible codons, and all occur in nature. However, there are not that many amino acids. In this project, you will explore various equivalence classes on the set of all codons and then compare the correspondence that occurs in nature.

For each of the following relations on codons, determine the number of equivalence classes, the size of each equivalence class, and describe the equivalence classes (you don’t necessarily need to fully write out each equivalence class).

1. The two codons have the same middle base (e.g., ACU ~ GCU).
2. The two codons have the same first two bases (e.g., ACC ~ ACU).
3. The two codons have the same number of A’s (e.g., AUA ~ GAA).
4. The two codons have the same number of C’s or U’s (e.g., ACU ~ GUU).
5. The two codons are made up of the same three bases (e.g., AUA ~ UAA).
6. The two codons use the same bases (e.g., AUA ~ UUA; this is different from #5).
7. In fact, the codons fall into 21 classes, indicating twenty basic amino acids and a signal to stop the genetic process called translation:
```alanine: {GCA, GCC, GCG, GCU}
arginine: {AGA, AGG, CGA, CGC, CGG, CGU}
asparagine: {AAC, AAU}
aspartic acid: {GAC, GAU}
cysteine: {UGC, UGU}
glutamic acid: {GAA, GAG}
glutamine: {CAA, CAG}
glycine: {GGA, GGC, GGG, GGU}
histidine: {CAC, CAU}
isoleucine: {AUA, AUC, AUU}
leucine: {CUA, CUC, CUG, CUU, UUA, UUG}
lysine: {AAA, AAG}
methionine: {AUG}
phenylalanine: {UUC, UUU}
proline: {CCA, CCC, CCG, CCU}
serine: {AGC, AGU, UCA, UCC, UCG, UCU}
threonine: {ACA, ACC, ACG, ACU}
tryptophan: {UGG}
tyrosine: {UAC, UAU}
valine: {GUA, GUC, GUG, GUU}
stop: {UAA, UAG, UGA}
```

This is not quite an easily described pattern; it is “close to” one of the previous equivalence relations. Which relation above is this close to?

```AAA: lysine
AAC: asparagine
AAG: lysine
AAU: asparagine

ACA: threonine
ACC: threonine
ACG: threonine
ACU: threonine

AGA: arginine
AGC: serine
AGG: arginine
AGU: serine

AUA: isoleucine
AUC: isoleucine
AUG: methionine
AUU: isoleucine

CAA: glutamine
CAC: histidine
CAG: glutamine
CAU: histidine

CCA: proline
CCC: proline
CCG: proline
CCU: proline

CGA: arginine
CGC: arginine
CGG: arginine
CGU: arginine

CUA: leucine
CUC: leucine
CUG: leucine
CUU: leucine

GAA: glutamic_acid
GAC: aspartic_acid
GAG: glutamic_acid
GAU: aspartic_acid

GCA: alanine
GCC: alanine
GCG: alanine
GCU: alanine

GGA: glycine
GGC: glycine
GGG: glycine
GGU: glycine

GUA: valine
GUC: valine
GUG: valine
GUU: valine

UAA: stop
UAC: tyrosine
UAG: stop
UAU: tyrosine

UCA: serine
UCC: serine
UCG: serine
UCU: serine

UGA: stop
UGC: cysteine
UGG: tryptophan
UGU: cysteine

UUA: leucine
UUC: phenylalanine
UUG: leucine
UUU: phenylalanine
```