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What does for all mean?

Looking for the meaning or definition of the word for all? Here's what it means.

Preposition
  1. In spite of, despite.
  2. (mathematics) (Indicates that a statement is true no matter what element of a set it is made about.)
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Examples
I was prepared for all this, and was most affable to the tormentor of a traveller's patience.
Still, for all of this, South Carolina is now represented in the U.S. Senate by Tim Scott, a Republican and an african-american.
Yet for all that, the hospitality of her little home in Ajaccio was lavish and famous.
The main law for all accentual versification is this, that verse-accent must always coincide with word-accent.
For this problem of the categories, in all its abstractness, is still a common problem for all of us.
Upward mobility and reward for work seems to be an afterthought for all concerned.

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