Write the Relation as a Set of Ordered Pairs | Relatively Prime Relation Write the Relation as a Set of Ordered Pairs Question Write the following relation as the set of ordered pairs: A relation \(R\) on the set \[ \{1,2,3,4,5,6,7\} \] defined by \[ (x,y)\in R \iff x \text{ is relatively prime to } […]

Write the Relation as a Set of Ordered Pairs | Relation Defined by x = 2y Write the Relation as a Set of Ordered Pairs Question Write the following relation as the set of ordered pairs: A relation \(R\) from the set \[ \{2,3,4,5,6\} \] to the set \[ \{1,2,3\} \] defined by \[ x=2y.

Find the Inverse Relation R⁻¹ | Relation Defined by y = x − 3 Find the Inverse Relation \(R^{-1}\) Question Find the inverse relation \[ R^{-1} \] if \(R\) is a relation from \[ \{11,12,13\} \] to \[ \{8,10,12\} \] defined by \[ y=x-3. \] Solution Using \[ y=x-3 \] For \[ x=11,\quad y=8 \]

Find the Inverse Relation R⁻¹ | Relation Defined by x + 2 = 8 Find the Inverse Relation \(R^{-1}\) Question Find the inverse relation \[ R^{-1} \] if \[ R=\{(x,y):x,y\in N,\ x+2=8\} \] Solution Given, \[ x+2=8 \] \[ x=6 \] Thus, \[ R=\{(6,y):y\in N\} \] Interchanging the coordinates, \[ R^{-1}=\{(y,6):y\in N\} \] Therefore, \[

Find the Inverse Relation R⁻¹ Find the Inverse Relation \(R^{-1}\) Question Find the inverse relation \[ R^{-1} \] if \[ R= \{ (1,2),(1,3),(2,3),(3,2),(5,6) \} \] Solution To find \[ R^{-1}, \] interchange the coordinates of each ordered pair. \[ (1,2)\rightarrow(2,1) \] \[ (1,3)\rightarrow(3,1) \] \[ (2,3)\rightarrow(3,2) \] \[ (3,2)\rightarrow(2,3) \] \[ (5,6)\rightarrow(6,5) \] Therefore, \[

Relation R on First Five Natural Numbers | Find R and R⁻¹ Relation \(R\) on the First Five Natural Numbers Where \(x\le y\) Question Let \[ A=\{1,2,3,4,5\} \] and let \(R\) be a relation on \(A\) defined as: \[ (x,y)\in R \iff x\le y \] Express \(R\) and \(R^{-1}\) as sets of ordered pairs. Determine

Relation from A to B Where x is Relatively Prime to y Relation from \(A\) to \(B\) Where \(x\) is Relatively Prime to \(y\) Question A relation \(R\) is defined from \[ A=\{2,3,4,5\} \] to \[ B=\{3,6,7,10\} \] as follows: \[ (x,y)\in R \iff x \text{ is relatively prime to } y. \] Express \(R\)

Which of the Following are Relations from A to B? Which of the Following are Relations from \(A\) to \(B\)? Question If \[ A=\{1,2,3\},\quad B=\{4,5,6\}, \] which of the following are relations from \(A\) to \(B\)? Give reasons. (i) \[ \{(1,6),(3,4),(5,2)\} \] (ii) \[ \{(1,5),(2,6),(3,4),(3,6)\} \] (iii) \[ \{(4,2),(4,3),(5,1)\} \] (iv) \[ A\times B \]

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, \[