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

To determine the nature of the relation \( R = \{(x, y) \in A \times A : x + y = 7\} \), we need to check for reflexivity, symmetry, and transitivity of this relation.

Set \( A \): \( \{1, 2, 3, 4, 5, 6, 7\} \)

Relation \( R \): Elements \( (x, y) \) such that \( x + y = 7 \)

1. Reflexivity: A relation is reflexive if for all elements \( x \in A \), \( (x, x) \in R \).

For reflexivity: Since \( x + x = 2x = 7 \) is not possible for any integer \( x \) in set \( A \), the relation is not reflexive.

2. Symmetry: A relation is symmetric if for all \( (x, y) \in R \), \( (y, x) \in R \).

Let's list the pairs in \( R \):

• (1, 6), because \( 1 + 6 = 7 \)
• (2, 5), because \( 2 + 5 = 7 \)
• (3, 4), because \( 3 + 4 = 7 \)
• (4, 3), because \( 4 + 3 = 7 \)
• (5, 2), because \( 5 + 2 = 7 \)
• (6, 1), because \( 6 + 1 = 7 \)

The pairs \( (3,4) \) and \( (4,3) \), \( (2,5) \) and \( (5,2) \), \( (1,6) \) and \( (6,1) \) indicate symmetry. For each pair \( (x, y) \) in \( R \), \( (y, x) \) is also in \( R \). Thus, the relation is symmetric.

3. Transitivity: A relation is transitive if for all \( (x, y) \in R \) and \( (y, z) \in R \), \( (x, z) \in R \).

Using the pairs in relation \( R \):

• From (1, 6) and (6, 1): Transitivity requires \( (1, 1) \) in \( R \), but \( 1 + 1 \neq 7 \).
• From (2, 5) and (5, 2): Transitivity requires \( (2, 2) \) and \( (5, 5) \), but neither satisfy \( x + x = 7 \).

No such connections hold without violating the relation's condition, so the relation is not transitive.

Conclusion: Based on the checks above, the relation \( R \) is symmetric but neither reflexive nor transitive.