Let P(S) denote the power set of S = {1, 2, 3 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) For relation $R_1$: The condition $(A \cap B^c) \cup (B \cap A^c) = \varnothing$ is the definition of the symmetric difference $A \Delta B = \varn

Vedclass · Accepted Answer

both $R_1$ and $R_2$ are equivalence relations

Vedclass · Answer

(A) For relation $R_1$: The condition $(A \cap B^c) \cup (B \cap A^c) = \varnothing$ is the definition of the symmetric difference $A \Delta B = \varnothing$,which implies $A = B$. Since $A = B$ is an equivalence relation (reflexive,symmetric,and transitive),$R_1$ is an equivalence relation.For relation $R_2$: The condition $A \cup B^c = B \cup A^c$ can be analyzed using set properties. $A \cup B^c = B \cup A^c \iff (A \cup B^c) \cap (A \cap B) = (B \cup A^c) \cap (A \cap B) \iff A = B$. Alternatively,using the Venn diagram regions where $a, b, c, d$ represent disjoint regions: $A = a \cup c$ and $B = b \cup c$. $A \cup B^c = (a \cup c) \cup (a \cup d) = a \cup c \cup d$. $B \cup A^c = (b \cup c) \cup (b \cup d) = b \cup c \cup d$. Equating these gives $a \cup c \cup d = b \cup c \cup d$,which implies $a = b$. Since $a$ and $b$ are the regions unique to $A$ and $B$ respectively,$a = b = \varnothing$ implies $A = B$. Thus,$R_2$ is also an equivalence relation.Therefore,both $R_1$ and $R_2$ are equivalence relations.

Let $P(S)$ denote the power set of $S = \{1, 2, 3, \ldots, 10\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $A R_1 B$ if $(A \cap B^c) \cup (B \cap A^c) = \varnothing$ and $A R_2 B$ if $A \cup B^c = B \cup A^c, \forall A, B \in P(S)$. Then:

Similar Questions

Given the relation $R = \{(1, 2), (2, 3)\}$ on the set $A = \{1, 2, 3\}$,the minimum number of ordered pairs which when added to $R$ make it an equivalence relation is

Let the relation $R_1$ be defined by $R_1 = \{ (a, b) | a \ge b, a, b \in R \}$. Then $R_1$ is:

Let $r$ be a relation on the set $N \times N$ defined by $(a, b)r(c, d) \Rightarrow a + d = b + c$. Then $r$ is:

If $R$ is the smallest equivalence relation on the set $\{1, 2, 3, 4\}$ such that $\{(1, 2), (1, 3)\} \subset R$,then the number of elements in $R$ is

The relation "congruence modulo $m$" is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module