Preliminaries Symmetric Dierence Set Dierence and Symmetric Dierence of Fuzzy Sets N.R. Vemuri A.S. Hareesh M.S. Srinath Department of Mathematics Indian Institute of Technology, Hyderabad and Department of Mathematics and Computer Science Sri Sathya Sai Institute of Higher Learning, India Fuzzy Sets Theory and Applications 2014, Liptovský Ján, Slovak Republic Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Outline 1 Preliminaries Introduction Earlier work 2 Symmetric Dierence Denition Examples Properties Applications Future Work References Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical set theory Set operations Union- [ Intersection - \ Complement - c Dierence -n Symmetric dierence - ∆ .... Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical set theory Set operations Union- [ Intersection - \ Complement - c Dierence -n Symmetric dierence - ∆ .... Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical set theory Set operations Union- [ Intersection - \ Complement - c Dierence -n Symmetric dierence - ∆ .... Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical set theory Set operations Union- [ Intersection - \ Complement - c Dierence -n Symmetric dierence - ∆ .... Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical set theory Set operations Union- [ Intersection - \ Complement - c Dierence -n Symmetric dierence - ∆ .... Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical set theory Set operations Union- [ Intersection - \ Complement - c Dierence -n Symmetric dierence - ∆ .... Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical vs Fuzzy operators Set operations Operation Classical Fuzzy Union [ t-conorm Intersection \ t-norm Complement c Fuzzy Negation Set dierence n ?? Symmetric dierence ∆ ?? Table : Classical vs Fuzzy operators Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical vs Fuzzy operators Set operations Operation Classical Fuzzy Union [ t-conorm Intersection \ t-norm Complement c Fuzzy Negation Set dierence n ?? Symmetric dierence ∆ ?? Table : Classical vs Fuzzy operators Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Classical vs Fuzzy operators Set operations Operation Classical Fuzzy Union [ t-conorm Intersection \ t-norm Complement c Fuzzy Negation Set dierence n ?? Symmetric dierence ∆ ?? Table : Classical vs Fuzzy operators Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dierence of (classical)sets Set dierence The set dierence of A; B is dened as A n B = fa 2 Aja 2= Bg Equivalent denition A n B = A \ Bc Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dierence of (classical)sets Set dierence The set dierence of A; B is dened as A n B = fa 2 Aja 2= Bg Equivalent denition A n B = A \ Bc Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dierence of (classical)sets Set dierence The set dierence of A; B is dened as A n B = fa 2 Aja 2= Bg Equivalent denition A n B = A \ Bc Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dierence of (classical)sets Set dierence The set dierence of A; B is dened as A n B = fa 2 Aja 2= Bg Equivalent denition A n B = A \ Bc Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dierence of (classical)sets Set dierence The set dierence of A; B is dened as A n B = fa 2 Aja 2= Bg Equivalent denition A n B = A \ Bc Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dierence of (classical)sets Set dierence The set dierence of A; B is dened as A n B = fa 2 Aja 2= Bg Equivalent denition A n B = A \ Bc Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Symmetric Dierence of (classical)sets Symmetric dierence The symmetric dierence of two sets A; B is dened as A∆B = (A n B) [ (B n A) Equivalent dention A∆B = (A \ Bc ) [ (B \ Ac ) Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Symmetric Dierence of (classical)sets Symmetric dierence The symmetric dierence of two sets A; B is dened as A∆B = (A n B) [ (B n A) Equivalent dention A∆B = (A \ Bc ) [ (B \ Ac ) Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Symmetric Dierence of (classical)sets Symmetric dierence The symmetric dierence of two sets A; B is dened as A∆B = (A n B) [ (B n A) Equivalent dention A∆B = (A \ Bc ) [ (B \ Ac ) Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Symmetric Dierence of (classical)sets Symmetric dierence The symmetric dierence of two sets A; B is dened as A∆B = (A n B) [ (B n A) Equivalent dention A∆B = (A \ Bc ) [ (B \ Ac ) Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Symmetric Dierence of (classical)sets Symmetric dierence The symmetric dierence of two sets A; B is dened as A∆B = (A n B) [ (B n A) Equivalent dention A∆B = (A \ Bc ) [ (B \ Ac ) Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Symmetric Dierence of (classical)sets Symmetric dierence The symmetric dierence of two sets A; B is dened as A∆B = (A n B) [ (B n A) Equivalent dention A∆B = (A \ Bc ) [ (B \ Ac ) Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Generalisations of Dierence operators Denition The dierence of two fuzzy sets can be dened as (1) Sd1 (x; y) = T (x; N(y)) (2) Sd2 (x; y) = x − T (x; y) where T ; N are a t-norm and a fuzzy negation, resp. Symmetric dierence of fuzzy sets ?? Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Generalisations of Dierence operators Denition The dierence of two fuzzy sets can be dened as (1) Sd1 (x; y) = T (x; N(y)) (2) Sd2 (x; y) = x − T (x; y) where T ; N are a t-norm and a fuzzy negation, resp. Symmetric dierence of fuzzy sets ?? Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Generalisations of Dierence operators Denition The dierence of two fuzzy sets can be dened as (1) Sd1 (x; y) = T (x; N(y)) (2) Sd2 (x; y) = x − T (x; y) where T ; N are a t-norm and a fuzzy negation, resp. Symmetric dierence of fuzzy sets ?? Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Generalisations of Dierence operators Denition The dierence of two fuzzy sets can be dened as (1) Sd1 (x; y) = T (x; N(y)) (2) Sd2 (x; y) = x − T (x; y) where T ; N are a t-norm and a fuzzy negation, resp. Symmetric dierence of fuzzy sets ?? Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Generalisations of Dierence operators Denition The dierence of two fuzzy sets can be dened as (1) Sd1 (x; y) = T (x; N(y)) (2) Sd2 (x; y) = x − T (x; y) where T ; N are a t-norm and a fuzzy negation, resp. Symmetric dierence of fuzzy sets ?? Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Generalisations of Dierence operators Denition The dierence of two fuzzy sets can be dened as (1) Sd1 (x; y) = T (x; N(y)) (2) Sd2 (x; y) = x − T (x; y) where T ; N are a t-norm and a fuzzy negation, resp. Symmetric dierence of fuzzy sets ?? Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Outline 1 Preliminaries Introduction Earlier work 2 Symmetric Dierence Denition Examples Properties Applications Future Work References Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dubois and Prade (1980) Two Examples S1(x; y) = jx − yj S2(x; y) = max(min(x; 1 − y); min(1 − x; y)) Note S1; S2 are only examples of fXoR operators No axiomatic denition was proposed. Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dubois and Prade (1980) Two Examples S1(x; y) = jx − yj S2(x; y) = max(min(x; 1 − y); min(1 − x; y)) Note S1; S2 are only examples of fXoR operators No axiomatic denition was proposed. Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dubois and Prade (1980) Two Examples S1(x; y) = jx − yj S2(x; y) = max(min(x; 1 − y); min(1 − x; y)) Note S1; S2 are only examples of fXoR operators No axiomatic denition was proposed. Vemuri, Sai Hareesh & Srinath Symmetric Dierence Preliminaries Introduction Symmetric Dierence Earlier work Dubois and Prade (1980) Two Examples S1(x; y) = jx − yj S2(x; y) = max(min(x; 1 − y); min(1 − x; y)) Note S1; S2 are only examples of fXoR operators