Prove "intersection distributes over union"
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The red

Transcribed Image Text:1 to which we will receive no satisfactory reply...
2each contained in some universe U, i.e. U is a set and X CUDY
3 each contained in some universe U, i.e. U is a set and X CUY
4each contained in some universe U, i.e. U is a set and X CU Ɔ Y and Z CU
5there is such a thing: Boolean algebra, see https://en.wikipedia.org/wiki/Boolean,lgebra
(I prefer to refer to it as Boolean arithmetic).

Transcribed Image Text:In lieu of asking what a set is, we should ask what we can do with sets. For
example, from two sets X and Y can we construct their (Cartesian) product:
XxY, and e.g. (the proof of) "the size of the product of two sets is the product
of their sizes" (has been requested for extra credit).
The union of sets X and Y, denoted XUY, is defined via
XUY:= {u € Uu € X or u € Y}
The intersection of sets X and Y³, denoted XnY, is defined via
XNY := {u € U\u € X and u € Y}
(2)
Prove "intersection distributes over union", i.e. for any three sets X, Y,
and Z1,
Xn(YUZ) = (XnY)u(Xnz).
So, in some "arithmetic of sets", there is a distributive property.
This suggests the following (purposefully incomplete) table of analogies:
Arithmetic Sets
+
(1)
1
(3)
Complete the above table of analogies now that we're aware of this
distributive property of the arithmetic of sets.
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