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2. (Counting Elements of Finite Sets). Let A, B, C be finite sets. Show that |AU BUC| = |A|+B+C|-|An B|-|AnC|-|BnC|+|AnBnC).

2. (Counting Elements of Finite Sets). Let A, B, C be finite sets. Show that |AU BUC| = |A|+B+C|-|An B|-|AnC|-|BnC|+|AnBnC).

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. (Finite Collections of Sets: Fundamental Products). Let be the universal set, and let U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 6, 7, 8), B = {1,4,5,6,9}, C = {3,4,5,6,9}. (i) Find all the fundamental products P₁ = An BnC, P₂ AnBnce, Ps= An Bºnc, P₁ = An Bence, Ps= An BnC, P = AnBnC, P = Aºn Benc, Ps= Aºn Benc of the sets A, B, C. (ii) Verify your answers by checking that the sets P are pairwise disjoint and their union is equal to U. It is allowed not to include the solution of the following problem into your submission. However, the result may be of considerable help when you will work on Problem 3, and so please try to prove it, and keep the proof for your records. If you are going to include the proof of the result in your submission, please make sure that the proof contains all necessary details and is typed correctly. 2. (Counting Elements of Finite Sets). Let A, B, C be finite sets. Show that |AU BUC| = |A|+|B| + |C| - |An B|-|AnC|-|BnC|+|AnBnC).
Transcribed Image Text:1. (Finite Collections of Sets: Fundamental Products). Let be the universal set, and let U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 6, 7, 8), B = {1,4,5,6,9}, C = {3,4,5,6,9}. (i) Find all the fundamental products P₁ = An BnC, P₂ AnBnce, Ps= An Bºnc, P₁ = An Bence, Ps= An BnC, P = AnBnC, P = Aºn Benc, Ps= Aºn Benc of the sets A, B, C. (ii) Verify your answers by checking that the sets P are pairwise disjoint and their union is equal to U. It is allowed not to include the solution of the following problem into your submission. However, the result may be of considerable help when you will work on Problem 3, and so please try to prove it, and keep the proof for your records. If you are going to include the proof of the result in your submission, please make sure that the proof contains all necessary details and is typed correctly. 2. (Counting Elements of Finite Sets). Let A, B, C be finite sets. Show that |AU BUC| = |A|+|B| + |C| - |An B|-|AnC|-|BnC|+|AnBnC).
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