Educational

If A×B ⊆ C×D and A×B ≠ Φ, Prove That A ⊆ C and B ⊆ D If \(A\times B \subseteq C\times D\) and \(A\times B \neq \Phi\), Prove That \(A \subseteq C\) and \(B \subseteq D\) Question If \[ A\times B \subseteq C\times D \] and \[ A\times B \neq \Phi, \] prove that […]

Prove That (A∩B)×C = (A×C)∩(B×C) Prove That \((A\cap B)\times C=(A\times C)\cap(B\times C)\) Question Prove that \[ (A\cap B)\times C=(A\times C)\cap(B\times C). \] Proof Let \[ (x,y)\in (A\cap B)\times C \] Then \[ x\in A\cap B \quad \text{and} \quad y\in C \] So, \[ x\in A,\quad x\in B \] and \[ y\in C \] Therefore, \[

Prove That (A∪B)×C = (A×C)∪(B×C) Prove That \((A\cup B)\times C=(A\times C)\cup(B\times C)\) Question Prove that \[ (A\cup B)\times C=(A\times C)\cup(B\times C). \] Proof Let \[ (x,y)\in (A\cup B)\times C \] Then \[ x\in A\cup B \quad \text{and} \quad y\in C \] So, \[ x\in A \quad \text{or} \quad x\in B \] If \[ x\in A

If A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, Find (A×B) ∪ (A×C) If \(A=\{1,2,3\}\), \(B=\{3,4\}\), \(C=\{4,5,6\}\), Find \((A\times B)\cup(A\times C)\) Question If \[ A=\{1,2,3\},\quad B=\{3,4\},\quad C=\{4,5,6\}, \] find \[ (A\times B)\cup(A\times C). \] Solution \[ A\times B= \{ (1,3),(1,4), (2,3),(2,4), (3,3),(3,4) \} \] \[ A\times C= \{ (1,4),(1,5),(1,6),

If A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, Find A×(B∪C) If \(A=\{1,2,3\}\), \(B=\{3,4\}\), \(C=\{4,5,6\}\), Find \(A\times(B\cup C)\) Question If \[ A=\{1,2,3\},\quad B=\{3,4\},\quad C=\{4,5,6\}, \] find \[ A\times(B\cup C). \] Solution \[ B\cup C=\{3,4,5,6\} \] \[ A\times(B\cup C) = \{ (1,3),(1,4),(1,5),(1,6), (2,3),(2,4),(2,5),(2,6), (3,3),(3,4),(3,5),(3,6) \} \] Therefore, \[ \boxed{ A\times(B\cup

If A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, Find (A×B) ∩ (A×C) If \(A=\{1,2,3\}\), \(B=\{3,4\}\), \(C=\{4,5,6\}\), Find \((A\times B)\cap(A\times C)\) Question If \[ A=\{1,2,3\},\quad B=\{3,4\},\quad C=\{4,5,6\}, \] find \[ (A\times B)\cap(A\times C). \] Solution \[ A\times B= \{ (1,3),(1,4), \] \[ (2,3),(2,4), \] \[ (3,3),(3,4) \} \] \[

If A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, Find A×(B∩C) If \(A=\{1,2,3\}\), \(B=\{3,4\}\), \(C=\{4,5,6\}\), Find \(A\times(B\cap C)\) Question If \[ A=\{1,2,3\},\quad B=\{3,4\},\quad C=\{4,5,6\}, \] find \[ A\times(B\cap C). \] Solution \[ B\cap C=\{4\} \] \[ A\times(B\cap C)=A\times\{4\} \] \[ = \{ (1,4),(2,4),(3,4) \} \] Therefore, \[ \boxed{ A\times(B\cap

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}, Verify That A×(B∩C) = (A×B)∩(A×C) Let \(A=\{1,2\}\), \(B=\{1,2,3,4\}\), \(C=\{5,6\}\), Verify That \(A\times(B\cap C)=(A\times B)\cap(A\times C)\) Question Let \[ A=\{1,2\},\quad B=\{1,2,3,4\},\quad C=\{5,6\} \] verify that \[ A\times(B\cap C)=(A\times B)\cap(A\times C). \] Solution \[ B\cap C=\phi \] \[ A\times(B\cap C)=A\times\phi=\phi \] \[

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6}, D = {5, 6, 7, 8}, Verify That A×C ⊂ B×D Let \(A=\{1,2\}\), \(B=\{1,2,3,4\}\), \(C=\{5,6\}\) and \(D=\{5,6,7,8\}\), Verify That \(A\times C \subset B\times D\) Question Let \[ A=\{1,2\},\quad B=\{1,2,3,4\}, \] \[ C=\{5,6\},\quad D=\{5,6,7,8\} \] Verify that \[ A\times C \subset

If A = {1, 2, 3}, B = {4}, C = {5}, Verify That A×(B − C) = (A×B) − (A×C) If \(A=\{1,2,3\}\), \(B=\{4\}\), \(C=\{5\}\), Verify That \(A\times(B-C)=(A\times B)-(A\times C)\) Question If \[ A=\{1,2,3\},\quad B=\{4\},\quad C=\{5\}, \] verify that \[ A\times(B-C)=(A\times B)-(A\times C). \] Solution \[ B-C=\{4\} \] \[ A\times(B-C)= \{ (1,4),(2,4),(3,4) \} \] \[