Let a set A - A1 - A2 - Ak- where Ai - Aj - - for i - j- 1 - i- j - k- Define the relation R from A to A by R - x- y - y - Ai if and only if x - Ai- 1 - i - k- Then- R is -

R={ (x, y): y ∈ A_i, iff x ∈ A_i, 1 ≤ i ≤ k } (1) Reflexive (a, a) ⇒ a ∈ A_i iff a ∈ A_i nbsp; (2) Symmetric (a, b) ⇒ a ∈A_i iff b ∈A_i (b, a) ∈ R as b ...

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Standard IX

Mathematics

Transitive Relations

Let a set A =...

Question

Let a set $A = A_{1} \cup A_{2} \cup \dots \cup A_{k}$ , where $A_{i} \cap A j = ϕ$ for $i \neq j, 1 \leq i, j \leq k$ . Define the relation $R$ from $A$ to $A$ by $R = {(x, y) : y \in A_{i} if and only if x \in A_{i}, 1 \leq i \leq k}$ . Then, $R$ is :

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Solution

$R = {(x, y) : y \in A_{i}, iff x \in A_{i}, 1 \leq i \leq k}$
(1) Reflexive
$(a, a) \Rightarrow a \in A_{i} iff a \in A_{i}$
(2) Symmetric
$(a, b) \Rightarrow a \in A_{i} iff b \in A_{i}$
$(b, a) \in R as b \in A_{i} iff a \in A_{i}$
(3) Transitive
$(a, b) \in R & (b, c) \in R$ .
$\Rightarrow a \in A_{i} iff b \in A_{i} & b \in A_{i} iff c \in A_{i}$
$\Rightarrow a \in A_{i} iff c \in A_{i}$
$\Rightarrow (a, c) \in R$ .
$\Rightarrow$ Relation is equivalence

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Let a set A=A1∪A2∪⋯∪Ak, where Ai∩Aj=ϕ for i≠j,1≤i,j≤k. Define the relation R from A to A by R={(x,y):y∈Ai if and only if x∈Ai,1≤i≤k}. Then, R is :

R={(x,y):y∈Ai, iff x∈Ai,1≤i≤k} (1) Reflexive (a,a)⇒a∈Ai iff a∈Ai (2) Symmetric (a,b)⇒a∈Ai iff b∈Ai (b,a)∈R as b∈Ai iff a∈Ai (3) Transitive (a,b)∈R & (b,c)∈R. ⇒a∈Ai iff b∈Ai & b∈Ai iff c∈Ai ⇒ a∈Ai iff c∈Ai ⇒ (a,c)∈R. ⇒ Relation is equivalence

Let a set $A = A_{1} \cup A_{2} \cup \dots \cup A_{k}$ , where $A_{i} \cap A j = ϕ$ for $i \neq j, 1 \leq i, j \leq k$ . Define the relation $R$ from $A$ to $A$ by $R = {(x, y) : y \in A_{i} if and only if x \in A_{i}, 1 \leq i \leq k}$ . Then, $R$ is :