Contents 1 Discrete structures F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 1 / 23 Discrete structures Section outline Lattices (contd.) Boolean lattice 1 Discrete structures Boolean lattice structure Sets Boolean algebra Relations Additional Boolean algebra Lattices properties F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 2 / 23 S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg Discrete structures Sets Sets A set A of elements: A = fa; b; cg F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B Universal set: U Empty set: ? = fg Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg S = fXjX 62 Xg S 2 S? [Russell’s paradox] F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 Set union: A [ B A B Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg S = fXjX 62 Xg S 2 S? [Russell’s paradox] F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 Set intersection: A \ B A B U Complement: S A Set difference: A − B = A \ B A B Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 U Complement: S A Set difference: A − B = A \ B A B Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 Set difference: A − B = A \ B A B Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 Discrete structures Sets Sets A set A of elements: A = fa; b; cg + Natural numbers: N = f0; 1; 2; 3;:::g or f1; 2; 3;:::g = Z Integers: Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g Universal set: U Empty set: ? = fg S = fXjX 62 Xg S 2 S? [Russell’s paradox] Set union: A [ B A B Set intersection: A \ B A B U Complement: S A F TECHN O OL E O T G U Y IT K T H S A N R I Set difference: A − B = A \ B A B A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 3 / 23 Complement of intersection (De Morgan): A \ B = A [ B U A B Power set of A: P(A) P(fa; bg) = f?; fag ; fbg ; fa; bgg Non-empty X1;:::; Xk is a partition of A if A = X1 [ ::: [ Xk and X{ \ X| = ? j{6=j A \ B, B \ A, A \ B and A [ B constitute a partition of U Discrete structures Sets Sets (contd.) Complement of union (De Morgan): A [ B = A \ B U A B F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 4 / 23 Power set of A: P(A) P(fa; bg) = f?; fag ; fbg ; fa; bgg Non-empty X1;:::; Xk is a partition of A if A = X1 [ ::: [ Xk and X{ \ X| = ? j{6=j A \ B, B \ A, A \ B and A [ B constitute a partition of U Discrete structures Sets Sets (contd.) Complement of union (De Morgan): A [ B = A \ B U A B Complement of intersection (De Morgan): A \ B = A [ B U A B F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 4 / 23 P(fa; bg) = f?; fag ; fbg ; fa; bgg Non-empty X1;:::; Xk is a partition of A if A = X1 [ ::: [ Xk and X{ \ X| = ? j{6=j A \ B, B \ A, A \ B and A [ B constitute a partition of U Discrete structures Sets Sets (contd.) Complement of union (De Morgan): A [ B = A \ B U A B Complement of intersection (De Morgan): A \ B = A [ B U A B Power set of A: P(A) F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 4 / 23 Non-empty X1;:::; Xk is a partition of A if A = X1 [ ::: [ Xk and X{ \ X| = ? j{6=j A \ B, B \ A, A \ B and A [ B constitute a partition of U Discrete structures Sets Sets (contd.) Complement of union (De Morgan): A [ B = A \ B U A B Complement of intersection (De Morgan): A \ B = A [ B U A B Power set of A: P(A) P(fa; bg) = f?; fag ; fbg ; fa; bgg F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 4 / 23 A \ B, B \ A, A \ B and A [ B constitute a partition of U Discrete structures Sets Sets (contd.) Complement of union (De Morgan): A [ B = A \ B U A B Complement of intersection (De Morgan): A \ B = A [ B U A B Power set of A: P(A) P(fa; bg) = f?; fag ; fbg ; fa; bgg Non-empty X1;:::; Xk is a partition of A if A = X1 [ ::: [ Xk and X{ \ X| = ? j{6=j F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 4 / 23 Discrete structures Sets Sets (contd.) Complement of union (De Morgan): A [ B = A \ B U A B Complement of intersection (De Morgan): A \ B = A [ B U A B Power set of A: P(A) P(fa; bg) = f?; fag ; fbg ; fa; bgg Non-empty X1;:::; Xk is a partition of A if A = X1 [ ::: [ Xk and X{ \ X| = ? j{6=j A \ B, B \ A, A \ B and A [ B constitute a partition of U F TECHN O OL E O T G U Y IT K T H S A N R I A N G A P I U D R N I 19 51 yog, km s kOflm^ Chittaranjan Mandal (IIT Kharagpur) SCLD January 21, 2020 4 / 23 Discrete structures Sets Set algebra Idempotence A [ A = A A \ A = A Associativity (A [ B) [ C = A [ (B [ C) (A \ B) [ C = A \ (B [ C) Commutativity A [ B = B [ A A \ B = B \ A Distributivity A [ (B \ C) = (A [ B) \ (A [ C) A \ (B [ C) = (A \ B) [ (A \ C) Identity A [ fg = A; A [ U = U A \ fg = fg ; A [ U = A Involution A¯ = A Complements U¯ = fg ; A [ A¯ = U fg¯ = U; A \ A¯ = fg DeMorgan A [ B = A¯ \ B¯ A \ B = A¯ [ B¯ F TECHN