MA265 Semana 2.2b Ejercicios Invariante

UPC – Departamento de Ciencias – Matemática Discreta (MA265) Profesores MA265 RESPUESTAS DE LOS EJERCICIOS PROPUESTOS 1 a) Dom (𝑅) = Ran (𝑅) = {1; 2; 3; 4; 5} b) 𝑀 𝑇 = ( 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 0) c) d) 𝒙 GI( 𝒙 ) GE( 𝒙 ) 1 3 3 2 1 2 3 3 3 4 2 2 5 2 1 e) 𝑇 2 = { (1; 1), (1; 3), (1; 4), (1; 5), (2,1), (2; 2), (2,3), (2; 5), (3; 1), (3; 2), (3; 3), (3; 5), (4; 1), (4; 4), (4; 5), (5; 1), (5; 4), (5; 5) } 𝑇 ∞ = 𝐴 × 𝐴 2 a) 𝑆 = { (1; 1), (1; 4), (2; 2), (2; 5), (3; 3), (3; 6), (4; 1), (4; 4), (5; 2), (5; 5), (6; 3), (6; 6) }, Dom (𝑅) = Ran (𝑅) = {1; 2; 3; 4; 5; 6} 4 3 2 5 1 UPC – Departamento de Ciencias – Matemática Discreta (MA265) Profesores MA265 b) 𝑀 𝑆 = ( 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1) c) a) b) 𝑀 𝑆 2 = 𝑀 𝑆 ∞ = ( 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1) 3 a) 𝑆 = {(1; 3), (2; 2), (3; 1), (3; 5), (4; 4), (5; 3)} , Dom (𝑆) = Ran (𝑆) = {1; 2; 3; 4; 5} b) 𝑀 𝑇 = ( 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0) 𝒙 GI( 𝒙 ) GE( 𝒙 ) 1 2 2 2 2 2 3 2 2 4 2 2 5 2 2 6 2 2 4 1 5 2 6 3 UPC – Departamento de Ciencias – Matemática Discreta (MA265) Profesores MA265 c) d) 𝒙 GI( 𝒙 ) GE( 𝒙 ) 1 1 1 2 1 1 3 2 2 4 1 1 5 1 1 e) 𝑆 3 = 𝑆 ∞ = 𝑆 = {(1; 3), (2; 2), (3; 1), (3; 5), (4; 4), (5; 3)} 4 a) 𝑆 = {(2; 2), (2; 4), (2; 6), (2; 12), (3; 3), (3; 6), (3; 12), (4; 4), (4; 12), (6; 6), (6; 12), (12; 12)}, Dom (𝑆) = Ran (𝑆) = {2; 3; 4; 6; 12} b) 𝑀 𝑆 = ( 1 0 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1) c) 3 12 6 4 2 3 2 5 1 4 UPC – Departamento de Ciencias – Matemática Discreta (MA265) Profesores MA265 d) e) 𝑀 𝑆 4 = 𝑀 𝑆 ∞ = ( 1 0 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1) 5 a) 𝑅 = { (𝑎 ; 𝑏) , (𝑎 ; 𝑐) , (𝑏 ; 𝑓) , (𝑐 ; 𝑑) , (𝑐 ; 𝑒) , (𝑑 ; 𝑏) , (𝑑 ; 𝑐) , (𝑒 ; 𝑓) , ( 𝑓 ; 𝑑 )} , Dom (𝑅) = {𝑎; 𝑏; 𝑐; 𝑑; 𝑒; 𝑓}, Ran (𝑅) = {𝑏; 𝑐; 𝑑; 𝑒; 𝑓} b) 𝑀 𝑆 = ( 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0) c) 𝒙 GI( 𝒙 ) GE( 𝒙 ) 𝑎 0 2 𝑏 2 1 𝑐 2 2 𝑑 2 2 𝑒 1 1 𝑓 2 1 d) 𝑅 2 = { (𝑎 ; 𝑑) , (𝑎 ; 𝑒) , (𝑎 ; 𝑓) , (𝑏 ; 𝑑) , (𝑐 , 𝑏) , (𝑐 ; 𝑐) , (𝑐 , 𝑓) , (𝑑 ; 𝑑) , ( 𝑑 ; 𝑒 ), ( 𝑑 ; 𝑓 ), ( 𝑒 ; 𝑓 ), ( 𝑓 ; 𝑏 ), ( 𝑓 ; 𝑐 )} 𝑅 ∞ = 𝐴 × 𝐴 − {(𝑎; 𝑎), (𝑏; 𝑎), (𝑐; 𝑎), (𝑑; 𝑎), (𝑒; 𝑎), (𝑓; 𝑎)} 6 a) 𝑅 = {(𝑎; 𝑎), (𝑎; 𝑏), (𝑎; 𝑐), (𝑎; 𝑒), (𝑏; 𝑐), (𝑐; 𝑐), (𝑑; 𝑏), (𝑑; 𝑐), (𝑑; 𝑑), (𝑑; 𝑒), (𝑒; 𝑐)} Dom (𝑅) = Ran (𝑅) = {𝑎; 𝑏; 𝑐; 𝑑; 𝑒} 𝒙 GI( 𝒙 ) GE( 𝒙 ) 2 1 4 3 1 3 4 2 2 6 3 2 12 5 1 UPC – Departamento de Ciencias – Matemática Discreta (MA265) Profesores MA265 b) 𝒙 GI( 𝒙 ) GE( 𝒙 ) 𝑎 1 4 𝑏 2 1 𝑐 5 1 𝑑 1 4 𝑒 2 1 c) 𝑀 𝑅 2 = 𝑀 𝑅 ∞ = ( 1 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0) 7 a) 𝑆 = { (𝑎 ; 𝑏) , (𝑎 ; 𝑓) , (𝑏 ; 𝑒) , (𝑐 ; 𝑒) , (𝑑 ; 𝑐) , (𝑑 ; 𝑓) , (𝑒 ; 𝑔) , (𝑓 ; 𝑐) , (𝑓 ; 𝑔) , ( 𝑔 ; 𝑑 )} Dom (𝑆) = {𝑎; 𝑏; 𝑐; 𝑑; 𝑒; 𝑓; 𝑔}, Ran (𝑆) = {𝑏; 𝑐; 𝑑; 𝑒; 𝑓; 𝑔} b) Relación 𝑅 Relación 𝑆 𝒙 GI( 𝒙 ) GE( 𝒙 ) 𝑎 3 7 𝑏 6 3 𝑐 3 3 𝑑 2 1 𝑒 4 4 𝑓 4 4 𝑔 4 4 𝒙 GI( 𝒙 ) GE( 𝒙 ) 𝑎 0 2 𝑏 1 1 𝑐 2 1 𝑑 1 2 𝑒 2 1 𝑓 2 2 𝑔 2 1 𝑐 d e 𝑏 𝑎 UPC – Departamento de Ciencias – Matemática Discreta (MA265) Profesores MA265 c) El par ordenado (𝑓, 𝑔) sí pertenece a 𝑅 3 d) El par ordenado (𝑑, 𝑎) no pertenece a 𝑅 ∞ 8 a) b) 𝑇 3 = { (1; 1), (1; 2), (1; 3), (1; 5), (1; 7), (2; 1), (2; 2), (2; 3), (2; 7), (3; 1), (3; 2), (3; 7), (4; 1), (4; 2), (4; 3), (4; 7), (5; 1), (5; 2), (5; 3), (5; 7), (6; 1), (6; 2), (6; 3), (6; 4), (6; 5), (6; 6)(6; 7), (7; 1), (7; 2), (7; 3), (7; 5), (7; 7) } y 𝑇 ∞ = { (1; 1), (1; 2), (1; 3), (1; 5), (1; 7), (2; 1), (2; 2), (2; 3), (2; 5), (2; 7), (3; 1), (3; 2), (3; 3), (3; 5), (3; 7), (4; 1), (4; 2), (4; 3), (4; 5), (4; 7), (5; 1), (5; 2), (5; 3), (5; 7), (6; 1), (6; 2), (6; 3), (6; 4), (6; 5), (6; 6)(6; 7), (7; 1), (7; 2), (7; 3), (7; 5), (7; 7) } 9.. a ) 𝑀 𝑅 = ( 1 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 ) 𝑀 𝑆 = ( 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 ) b) c) 𝑀 𝑅 2 = ( 1 0 0 1 0…