Ribonucleic acid "codons" are strings of 3 ordered nucleotide bases in which each nucleotide is one of adenine (A), cytosine (C), guanine (G), or uracil (U). Each codon "signals" or encodes an amino acid. Suppose two codons XYZ and PQR are - related, and we write XYZ - PQR, if they have the same three distinct kinds of bases. Thus, for example CUG and UGC are - related, written CUG - UGC, since they both have one cytosine C, one guanine G, and one uracil U; however CUG is not - related to CGA, since both have a C and a G, but one also has an A and the other has a U. If - is an equivalence relation on the set of 3 nucleotide codons with 3 different bases, then prove it is, and determine how many equivalence classes there are. If it is not an equivalence relation then describe which of the three rules it fails to satisfy and which it satisfies. In either case give detailed explanations of your answers. Assume that order matters, that is CUG is not the same codon as GUC.

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Chapter2: Second-order Linear Odes
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Ribonucleic acid "codons" are strings of 3 ordered nucleotide bases in which each nucleotide is
one of adenine (A), cytosine (C), guanine (G), or uracil (U). Each codon "signals" or encodes an amino
acid. Suppose two codons XYZ and PQR are - related, and we write XYZ - PQR, if they have the
same three distinct kinds of bases. Thus, for example CUG and UGC are - related, written CUG -
UGC, since they both have one cytosine C, one guanine G, and one uracil U; however CUG is not -
related to CGA, since both have a C and a G, but one also has an A and the other has a U.
If - is an equivalence relation on the set of 3 nucleotide codons with 3 different bases, then prove it
is, and determine how many equivalence classes there are. If it is not an equivalence relation then
describe which of the three rules it fails to satisfy and which it satisfies. In either case give detailed
explanations of your answers. Assume that order matters, that is CUG is not the same codon as
GUC.
Transcribed Image Text:Ribonucleic acid "codons" are strings of 3 ordered nucleotide bases in which each nucleotide is one of adenine (A), cytosine (C), guanine (G), or uracil (U). Each codon "signals" or encodes an amino acid. Suppose two codons XYZ and PQR are - related, and we write XYZ - PQR, if they have the same three distinct kinds of bases. Thus, for example CUG and UGC are - related, written CUG - UGC, since they both have one cytosine C, one guanine G, and one uracil U; however CUG is not - related to CGA, since both have a C and a G, but one also has an A and the other has a U. If - is an equivalence relation on the set of 3 nucleotide codons with 3 different bases, then prove it is, and determine how many equivalence classes there are. If it is not an equivalence relation then describe which of the three rules it fails to satisfy and which it satisfies. In either case give detailed explanations of your answers. Assume that order matters, that is CUG is not the same codon as GUC.
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