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Consider $A \subset \{0,1\}^n$

I want $A$ to have $2$ properties.

$1.$ $A$ is increasing, i.e., If $x \in A$ and $x \subset y$ then $y \in A$ too.

$2.$ $A^c$ is equal to set $B=\{x | x^c \in A\}$

Is there any characterization for such a set? I have to example for it. But I want to find an IFF condition for such sets...

$e1)$ $A=\{x|$ first coordinate of $x$ is $1\}$

$e2)$ Fix an odd number of coordinates. $A= \{x| x$ contains at least half of coordinates equal to $1\}$ [For even number there is a similar example]

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    how is $x\subset y$ defined? – 2017-02-08
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    Note that your second example only works if $n$ is odd. – 2017-02-08
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    Do you identify $\{ 0, 1 \}^n$ with the powerset of $\{1, 2, \dotsc, n\}$? – 2017-02-08
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    Since you haven't responded to user160738's question yet, let me be more specific. By the definition you gave, elements of $A$ are $n$-tuples of values of $0$ or $1$. They are not sets, so the notation $x \subset y$ is not defined for them by standard convention. There is of course the natural identification of such $n$-tuples with subsets of $\{1,2,...,n\}$, but if you are making use of this, you should say so. – 2017-02-09
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    @PaulSinclair Yeah. But I supposed to it is not necessary.... – 2017-02-09
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    @EricWofsey No. For even numbers you can consider half of middle sets so that there was not complements – 2017-02-09
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    @Axolotl: Yes, you can find a similar example that works, but that's not what you wrote before. – 2017-02-09
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    @EricWofsey Yeah. But it was not the matter to emphasize. I'm looking for all possibilities. – 2017-02-09
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    @EricWofsey I found a generalization for second example – 2017-02-10
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    MO copy of the question: [The set of complements equal to the complement of set](https://mathoverflow.net/q/261847) – 2017-12-11

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