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In definition of transitive dependency we use conditions as

1. A → B
2. It is not the case that B → A
3. B → C

I am not able to understand why we need condition 2 for this ?
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Transitive dependency occurs when there is FD from non prime attribute to non prime attribute.

For given A->B and B->C A is candidate key hence prime attribute and B->C is Non-prime attribute to Non -Prime attribute but if B->A is there then B is also candidate key hence Prime Attribute.In this case FD B->C is from PA to NPA which is the case of partial dependency.
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Transitive dependency occurs when there is FD from non prime attribute to non prime attribute.

false, you are assuming it is 3 NF, it is not by default.

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@Shaik

Can you elaborate I'm unable to get you....3NF occurs when there's no transitive dependency and Transitive dependency is NPA to NPA..Isn't it?? Please explain where I'm wrong!
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transitive dependency mean, A-> B, B->C

those may or may not prime attributes....

in 3NF, only prime transitivity allowed... But in the question it is not mentioned, took 3NF.

But in your previous comment, you mentioned, transitivity means non-prime to non-prime.

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Shaik transitive dependency occurs when a non prime attribute is transitively dependent on the key right?

So if B->A is a valid FD in a relation R (A,B,C) where the FDs A->B and B->C also hold, then both A and B become candidate keys for the relation R and so B->C is no longer a transitive dependency right?

A functional dependency X->Y is said to be in 3NF if either X is a superkey or Y is a prime attribute.

So following this above definition, B->C no longer violates this rule as B is a superkey if B->A hold on relation R.

But in the absence of B->A, then neither B is a superkey nor C is a prime attribute. So the relation R wont be in 3NF then.

am i right Shaik?

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if the question was about 3NF, then you are right !

but note that

transitive dependency occurs when a non prime attribute is transitively dependent on the key right?

transitive dependency can be prime transitivity or non-prime transitivity

A --> B and B ---> C then A --> C is a transitivity

A --> B, B--->A and B ---> C then also A ---> C is a transitivity.

But

B --> A and B --->C doesn't lead to transitivity ( this is the meaning in the question)

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Shaik can u pls explain what u mean by prime transitivity and non prime transitivity?

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A --> B and B ---> C then A --> C is a transitivity

this is non-prime transitivity right?since prime attribute A is being able to derive C through a non prime attribute B.

A --> B, B--->A and B ---> C then also A ---> C is a transitivity.

this is prime transitivity right? since prime attribute A is being able to determine C through prime attribute B.

Is this the concept Shaik?

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Prime transitivity means

The transitive dependency is between two prime attributes...

Non-prime transitivity means

The transitive dependency is between two non-prime attributes
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Consider a relation R(A,B,C) where the following FDs hold:

A->B, B->C, C->A, C->B

Here A and C both are candidate keys and hence prime attributes.

According to your definition of prime transitivity, A->C is a prime transitivity right?

A functional dependency X->Y is said to be in 3NF if either X is a superkey or Y is a prime attribute.

But from this defintion of 3NF above, the relation is in 3NF since every FD satisfies this requirement right?

So whats the use of this prime transitivity?

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In X -> Y, Y is PA and X is not key, then it means it is prime transitivity due to X should be a prime attributes..

Let A-> B, B->C and C ->D, in this B->D is non-prime transitivity.
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Shaik but we allow prime transitivity in 3NF right? prime transitivity will be removed in BCNF right?

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i didn't comment anything about 3NF or BCNF....

i just comment about transitivity even in the question they are interested in transitive dependency but not for any Normall form... just you people make it complicate :)
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@shaik​​​​

you told A --> B, B--->A and B ---> C then also A ---> C is a transitivity.

but this is not matching with the defination in the question
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the question saying, B -> A, B -> C then A ---> C is not transitive.
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@shaik

but the question is A → B,  It is not the case that B → A   ,B → C

then A->C transitive dependency

now my question is what is the need of not B->A

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his meaning,

A --> B but it is not B--> A , B --> C