MATEMÁTICA

The powerset of a set is a Boolean ring under intersection and symmetric difference

Let [pic 1] be a nonempty set and let [pic 2] denote the power set of [pic 3]. Define operators [pic 4] and [pic 5] on [pic 6] as follows: [pic 7] and [pic 8].

1. Prove that [pic 9] is a ring under these operators.
2. Prove that this ring is commutative, has an identity, and is Boolean.

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Before we begin, note that for all sets [pic 10], [pic 11], and [pic 12], the following hold.

• [pic 13]
• [pic 14]
• [pic 15]
• [pic 16]

Now let [pic 17]. We have the following.

[pic 18]	=	[pic 19]
	=	[pic 20] [pic 21]
	=	[pic 22] [pic 23]
	=	[pic 24] [pic 25]
	=	[pic 26] [pic 27] [pic 28] [pic 29] [pic 30] [pic 31]
	=	[pic 32] [pic 33] [pic 34] [pic 35]
	=	[pic 36] [pic 37] [pic 38] [pic 39] [pic 40] [pic 41]
	=	[pic 42] [pic 43] [pic 44]
	=	[pic 45] [pic 46]
	=	[pic 47] [pic 48]
	=	[pic 49] [pic 50]
	=	[pic 51] [pic 52]
	=	[pic 53] [pic 54]
	=	[pic 55]

So [pic 56] is associative. Moreover,

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