Dependent and Independent Components

Florentin Smarandache

Taking into consideration the relationships vertices (edges and hyper-edges), then between n-supervertices, we construct n-HyperEdges.

So, the whole graph is an n-SuperHyperGraph.

Dependent and Independent Components

Florentin Smarandache to Mary Jansi

Dependent components means when the components influence each other, i.e. if one changes its value the others change too.

Independent components means when the components do not influence each other, i.e. if one component changes its value it is not necessarily for the other components to change. Real life examples 1) Suppose you watch a football match.

You have: T = chance that your team wins, I = chance of tie game, and F = chance that your team loses.

You are a mathematician and you know that the sum of all space probabilities is 1. So: T + I + F = 1.

Herein, all T, I, F are dependent on each other.

For example, if T = 0.6, I = 0.1, F = 0.3, you have T + I + F = 0.6+0.1+0.3 = 1.

But let's say, you change your mind and you predict that the chance of winning is T = 0.7; then mandatory I and/or F should change their values since the sum has to still be 1.

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Nidus Idearum. Scilogs, IX: neutrosophia perennis

2) Suppose three people, which do not communicate with each other, and they stay in places far from each other.

The first one John predicts the chance that your team wins is T = 0.8.

The second one George predicts the chance that the match is tie I = 0.3.

The third one predicts the chance that your team loses is F = 0.5.

Clearly T, I, F are independent, since the three people do not communicate with each other and do not stay together, so they think independently of each other.

Now, if you sum T+I+F you cannot get 1, only in extremely rare coincidences.

Suppose now that John changes his prediction and says that T = 0.7, but he does not say anything to the other two.

The other two may change or may not.

That's why in a neutrosophic set/logic/probability we have T+I+F ≤ 3, and each one T, I, F can be any number in [0, 1]. T I, F are independent.

In intuitionistic fuzzy set, T+I+F = 1, so T, I (called “hesitancy” in this case), F are dependent. 3) Suppose T and F are dependent. Then T and F depend on each other, or T+F = 1 (as in fuzzy logic). Another real life example

Suppose you strongly believe that in the above football game one of the teams will win, so you do not believe in a tie game.