Question

Relation \( R \) on \( A = \{1,2,3,4,5,6,7\} \):
\[ R = \{(a,b) : a \text{ and } b \text{ are both odd or both even} \} \] Show that \( R \) is an equivalence relation and find the equivalence class \([1]\).

Answer 1

Step 1: Reflexivity
For all \( a \in A \), \( a \) and \( a \) are either both odd or both even, so \( (a, a) \in R \).
⇒ \( R \) is reflexive.
Step 2: Symmetry
Suppose \( (a, b) \in R \). That means \( a \) and \( b \) are both odd or both even.
So, \( b \) and \( a \) are also both odd or both even ⇒ \( (b, a) \in R \).
⇒ \( R \) is symmetric.
Step 3: Transitivity
Suppose \( (a, b) \in R \) and \( (b, c) \in R \).
That means \( a \) and \( b \) have the same parity, and \( b \) and \( c \) have the same parity.
⇒ \( a \) and \( c \) also have the same parity ⇒ \( (a, c) \in R \).
⇒ \( R \) is transitive.
Hence, \( R \) is an equivalence relation.
Step 4: Find [1]
\([1] = \{ b \in A : (1, b) \in R \}\) ⇒ \( b \) must have the same parity as 1 (i.e., odd).
Odd elements in \( A \) = \( \{1, 3, 5, 7\} \)
So, \( [1] = \{1, 3, 5, 7\} \)