In this sentence what does 所 mean？ Does it mean completely?
In this sentence what does 所 mean？欧盟则是一个由独立的民主国家所组成的同盟， does it mean completely?

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it indicates passive voice, it can be omitted sometimes
for ex: 
欧盟由XX国家所组成 = XX国家组成欧盟 EU is made up of several countries
这辆车由我所拥有 = 我拥有这辆车 The car is owned by me

Answer

http://www.italki.com/answers/question/161439.htm

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For example, 白色是我所喜欢的颜色。

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所is more like "from" 
The European nation is made from ·····

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No, it should be a structural particle. 结构助词放在词、词组的前面或后面，用以表示句子的结构关系。常见的结构助词有前缀的“所”，以及后附的“的、地、得、似的”等等。