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 \[(A \cap B^c) \cup (B \cap A^c) = ,\]and \(A R_2 B\) if\[A \cup B^c = B \cup A^c,\]for all \(A, B \in P(S)\). Then:
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 \[(A \cap B^c) \cup (B \cap A^c) = ,\]and \(A R_2 B\) if\[A \cup B^c = B \cup A^c,\]for all \(A, B \in P(S)\). Then:
both\(R_1\) and \(R_2\) are equivalence relations
only\(R_2\) is an equivalence relation
only \(R_1\) is an equivalence relation
both \(R_1\)and \(R_2\) are not equivalence relations
The Correct Option is A
Solution and Explanation
Let \(S = \{1, 2, 3, \ldots, 10\}\), and \(P(S)\) denote the power set of \(S\).
Relation \(R_1: A R_1 B\) if
\[(A \cap B^c) \cup (B \cap A^c) = \varnothing.\]
This implies \(A = B\).
Step 1: Check Reflexivity
For any \(A \in P(S)\),
\[(A \cap A^c) \cup (A^c \cap A) = \varnothing.\]
Thus, \(AR_1A\). Therefore, \(R_1\) is reflexive.
Step 2: Check Symmetry
If \(AR_1B\), then \((A \cap B^c) \cup (B \cap A^c) = \varnothing\). This implies \(A = B\), and thus \(BR_1A\). Therefore, \(R_1\) is symmetric.
Step 3: Check Transitivity
If \(AR_1B\) and \(BR_1C\), then \(A = B\) and \(B = C\), which implies \(A = C\). Therefore, \(R_1\) is transitive.
Thus, \(R_1\) is an equivalence relation.
Relation \(R_2\): \(A R_2 B\) if
\[A \cup B^c = B \cup A^c.\]
Step 1: Check Reflexivity
For any \(A \in P(S)\),
\[A \cup A^c = A \cup A^c.\]
Thus, \(AR_2A\). Therefore, \(R_2\) is reflexive.
Step 2: Check Symmetry
If \(AR_2B\), then \(A \cup B^c = B \cup A^c\). By symmetry of the union operation, \(B \cup A^c = A \cup B^c\), which implies \(BR_2A\). Therefore, \(R_2\) is symmetric.
Step 3: Check Transitivity
If \(AR_2B\) and \(BR_2C\), then:
\[A \cup B^c = B \cup A^c,\]
\[B \cup C^c = C \cup B^c.\]
Using substitution, we get:
\[A \cup C^c = C \cup A^c,\]
which implies \(AR_2C\). Therefore, \(R_2\) is transitive.
Thus, \(R_2\) is an equivalence relation.
Conclusion: Both \(R_1\) and \(R_2\) are equivalence relations
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".