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Prove "intersection distributes over union"

Prove "intersection distributes over union"

Advanced Engineering Mathematics
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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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: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).
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.
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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