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2.12 Prove the following equation. 0< H j®B (x) < 0.5 where AOB = (AOB)U(ANB) that is, is simple disjunctive sum operator. Claim: 0<4493 (x)<0.5. Proof when membership values of x in both A and B lie between 0 and 0.5. The other cases are dealt with in identical fashion. When 0

2.12 Prove the following equation. 0< H j®B (x) < 0.5 where AOB = (AOB)U(ANB) that is, is simple disjunctive sum operator. Claim: 0<4493 (x)<0.5. Proof when membership values of x in both A and B lie between 0 and 0.5. The other cases are dealt with in identical fashion. When 0

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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2.12 Prove the following equation. 0< H j®B (x) < 0.5 where AOB = (AOB)U(ANB) that is, is simple disjunctive sum operator.
Transcribed Image Text:2.12 Prove the following equation. 0< H j®B (x) < 0.5 where AOB = (AOB)U(ANB) that is, is simple disjunctive sum operator.
Claim: 0<4493 (x)<0.5. Proof when membership values of x in both A and B lie between 0 and 0.5. The other cases are dealt with in identical fashion. When 0<u,(x)<0.5 and 0<µ, (x)<0.5, then following cases hold. 0.5 s Min(1– 4, (x))<1 and 0.5 <Min(1- 43(x))<l This implies, 0s Min(1- 4,(x).4(x)) <0.5 and 0<Min(4,(x),1– 43 (x))<0.5. Hence, Мах (Min (и, (х).1-н, (х)). Min(1- u.(*).и, (х))} will lhies between 0 and 0.5. That is, 0<µe3 (x)<0.5 by result (6). (Proved)
Transcribed Image Text:Claim: 0<4493 (x)<0.5. Proof when membership values of x in both A and B lie between 0 and 0.5. The other cases are dealt with in identical fashion. When 0<u,(x)<0.5 and 0<µ, (x)<0.5, then following cases hold. 0.5 s Min(1– 4, (x))<1 and 0.5 <Min(1- 43(x))<l This implies, 0s Min(1- 4,(x).4(x)) <0.5 and 0<Min(4,(x),1– 43 (x))<0.5. Hence, Мах (Min (и, (х).1-н, (х)). Min(1- u.(*).и, (х))} will lhies between 0 and 0.5. That is, 0<µe3 (x)<0.5 by result (6). (Proved)
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