Question

Let \( A = \{1, 2, 3, 4, 5, 6, 7\} \). Then the relation \( R = \{(x, y) \in A \times A : x + y = 7\} \) is:

• A. Symmetric but neither reflexive nor transitive
• B. Transitive but neither symmetric nor reflexive
• C. An equivalence relation
• D. Reflexive but neither symmetric nor transitive

Hint:

Check symmetry by verifying if \( (x, y) \in R \implies (y, x) \in R \). For reflexivity, \( (x, x) \in R \) must hold, and for transitivity, validate the chain condition.

Answer 1

The Correct Option is A

The relation \( R \) is defined as:

\[ R = \{(x, y) \in A \times A : x + y = 7\}. \]

By substitution, the pairs in \( R \) are:

\[ R = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\}. \]

1. Symmetry:
   • For \( (x, y) \in R \), \( x + y = 7 \).
   • Therefore, \( (y, x) \in R \) since \( y + x = 7 \).
   • Hence, \( R \) is symmetric.
2. Reflexivity:
   • For \( R \) to be reflexive, \( (x, x) \in R \) for all \( x \in A \).
   • This is not true because \( x + x \neq 7 \) for any \( x \in A \).
   • Hence, \( R \) is not reflexive.
3. Transitivity:
   • For \( R \) to be transitive, if \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \).
   • However, this is not satisfied as there are no such \( x, y, z \) in \( R \) to form a valid chain.
   • Hence, \( R \) is not transitive.

Thus, \( R \) is symmetric but neither reflexive nor transitive.