IIT JEE Main Maths -Unit 1 -Relation - Exam Style Questions- New Syllabus
▶️ Answer/Explanation
The number of non-empty equivalence relations on a set with \( 3 \) elements is equal to the number of partitions of the set. For the set \( \{1,2,3\} \), the partitions are:
\(1.\ \{\{1,2,3\}\}\)
\(2.\ \{\{1\}, \{2,3\}\}\)
\(3.\ \{\{2\}, \{1,3\}\}\)
\(4.\ \{\{3\}, \{1,2\}\}\)
\(5.\ \{\{1\}, \{2\}, \{3\}\}\)
Thus, there are \( 5 \) non-empty equivalence relations. The correct answer is \((3)\ 5\).
▶ Answer/Explanation
Ans. (3)
Sol. Transitivity
$(1, 2) \in R$, $(2, 3) \in R$ $\Rightarrow (1, 3) \in R$
For reflexive $(1, 1)$, $(2, 2)$, $(3, 3) \in R$
Now $(2, 1)$, $(3, 2)$, $(3, 1)$
$(3, 1)$ cannot be taken
(1) $(2, 1)$ taken and $(3, 2)$ not taken
(2) $(3, 2)$ taken and $(2, 1)$ not taken
(3) Both not taken
therefore
Sol. Transitivity
$(1, 2) \in R$, $(2, 3) \in R$ $\Rightarrow (1, 3) \in R$
For reflexive $(1, 1)$, $(2, 2)$, $(3, 3) \in R$
Now $(2, 1)$, $(3, 2)$, $(3, 1)$
$(3, 1)$ cannot be taken
(1) $(2, 1)$ taken and $(3, 2)$ not taken
(2) $(3, 2)$ taken and $(2, 1)$ not taken
(3) Both not taken
therefore
▶️ Answer/Explanation
Ans. (4)
Sol. \(A = \{1, 2, 3, 4\}\)
For relation to be reflexive
\(R = \{(1,2), (2, 3), (3,3)\}\)
Minimum elements added will be
\((1,1), (2,2), (4,4) (2,1) (3,2) (3,2) (3,1) (1,3)\)
\(\therefore\) Minimum number of elements = 7
Option : (4)