Algebraic Dependence of (x,z): If (x,y,z) & (x,y) Dependent

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In summary, when (x,y,z) and (x,y) are dependent, it means there is a relationship between the two sets of data. Algebra can be used to determine this relationship by setting up equations and solving for variables. It is possible for (x,z) to be dependent even if (x,y,z) and (x,y) are not. Algebraic dependence can be applied to real-world situations to model and predict relationships between variables. The difference between algebraic dependence and independence is that the former has a relationship between two sets of data, while the latter does not.
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disregardthat
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If (x,y,z) and (x,y) is algebraically dependent, is then (x,z) algebraically dependent?

Does by the way anyone have a good web-source for information about transcendence degrees?
 
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Definitions needed. What are x,y,z? What does it mean for an ordered triple and and ordered pair to be algebraically dependent?
 
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Any subset of an algebraically independent set is algebraically independent.
 

1. What does it mean for (x,y,z) and (x,y) to be dependent?

When two sets of data are dependent, it means that there is a relationship between the two sets. In this case, the values of (x,y) can be used to determine the value of (x,y,z).

2. How is algebra used to determine the dependence of (x,y,z) and (x,y)?

Algebra can be used to find the relationship between (x,y,z) and (x,y) by setting up equations and solving for the variables. If the equations have the same solution, then the sets are dependent.

3. Can (x,z) be dependent even if (x,y,z) and (x,y) are not dependent?

Yes, it is possible for (x,z) to be dependent while (x,y,z) and (x,y) are not dependent. This means that there is a relationship between (x,z) that is not influenced by the relationship between (x,y,z) and (x,y).

4. How does the concept of algebraic dependence apply to real-world situations?

In real-world situations, algebraic dependence can be used to model and predict relationships between different variables. For example, it can be used in financial analysis to determine how changes in one variable will affect another variable.

5. What is the difference between algebraic dependence and algebraic independence?

Algebraic dependence means that there is a relationship between two sets of data, while algebraic independence means that there is no relationship between the two sets. In other words, the values in one set do not affect the values in the other set.

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