The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is __________.

To find the number of symmetric relations on the set {1, 2, 3, 4} that are not reflexive, we first need to understand a few concepts.

Understanding Symmetric Relations

A symmetric relation on a set means that if (a, b) is in the relation, then (b, a) must also be in the relation. For a set with n elements, the total number of possible pairs (including reflexive pairs) is given by:

• Reflexive pairs: (1,1), (2,2), (3,3), (4,4)
• Non-reflexive pairs: (1,2), (1,3), (1,4), (2,3), (2,4), (3,4)

Total Pairs Calculation

For the set {1, 2, 3, 4}, there are:

• 4 reflexive pairs
• 6 non-reflexive pairs

Counting All Symmetric Relations

Each non-reflexive pair can either be included in the relation or not, and since each pair is symmetric, we only consider one direction. Thus, for the 6 non-reflexive pairs, we have:

Number of ways to choose pairs = 2⁶ = 64

Including the reflexive pairs, the total number of symmetric relations is:

Number of symmetric relations = 2⁴ * 2⁶ = 16 * 64 = 1024

Excluding Reflexive Relations

Now, we need to find the number of symmetric relations that are reflexive. A reflexive relation must include all reflexive pairs, so we only consider the non-reflexive pairs:

Number of reflexive symmetric relations = 2⁶ = 64

Final Calculation

To find the number of symmetric relations that are not reflexive, we subtract the number of reflexive relations from the total number of symmetric relations:

Number of non-reflexive symmetric relations = Total symmetric relations - Reflexive symmetric relations

Number of non-reflexive symmetric relations = 1024 - 64 = 960

Thus, the number of symmetric relations defined on the set {1, 2, 3, 4} that are not reflexive is 960.