The verb "ser" is used in these sentences for the following reasons:
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Ella es mi madre. - This sentence uses "ser" to indicate a permanent relationship (identity) between "Ella" (she) and "mi madre" (my mother).
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Yo soy una profesora. - Here, "ser" is used to describe a permanent characteristic or identity of the speaker (I am a teacher).
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Son las tres. - In this case, "ser" is used to express time, which is a state that is typically considered permanent.
Overall, "ser" is used for identity, occupation, origin, and time, which are considered essential or defining characteristics.