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If i'm understanding your question correctly you're wondering what the purpose for Mendel's pure line studies. AABB, AAbb, aaBB, aabb are all pure line that Mendel worked with.

Remember this is a time where genetics has not yet been founded officially. When Mendel tended the gardens he noticed some rose plants would be red, some white. He also noticed that in pea plants some pea seeds would be round and yellow, round and green, yellow and wrinkled, and green and wrinkled.

Mendel wanted to know what was causing these events occur so he used plants that were always being red, always being white (AA(red) x aa(white)) and these were pure lines. To make sense of all this he made 4 proposals.

He made 4 proposals: Key:(Alleles(UNITS) genes(FACTORS)

1. Unit Factors come in pairs: Notice how unit and factors are together, this is because he is referring them together.

Unit factors(genes and alleles) come in pairs: example AABB: AA(red)xBB(tall)

there are always 2 alleles for each gene: one allele for each parent

2. Dominance/recessiveness:

When the units for a particular factor, or trait, are different, one unit is dominant over the other. Example: (Aa) A is dominant over a. AA or aa express no dominance over the other that is why they must be different.

3. Segregation: This is where Punnett Square was developed from. This is also where genotype and phenotype ratios started.

The unit factors segregate randomly during gamete formation so each gamete has equal likelihood of receiving either unit of a given factor. Meaning if you cross two pure lines together all your progeny will be the same. example AABB x aabb will create one progeny genotype: AaBb

4. Independent Assortment: This one has exceptions that was later discovered like Epistasis, lethal alleles, codominance, incomplete dominance etc.

The units of different factors segregate into the gametes independently of one another. If you were to draw a Punnett Square and cross together this is based on assumption that all segregate independently.

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