Counting Anti-Symmetric Relations on a Set with Specific Ordered Pairs

In the realm of discrete mathematics, understanding the number of anti-symmetric relations on a set with given ordered pairs is a fundamental problem. This article provides a detailed explanation of how to solve such a problem, leveraging the principles of set theory and combinatorics. We will explore the steps to find the number of anti-symmetric relations on the set (A {1, 2, 3, 4, 5, 6}) that contain the ordered pairs (2, 2), (3, 4), and (5, 6).

Understanding Anti-Symmetric Relations

An anti-symmetric relation (R) on a set (A) satisfies a specific condition: if both (a , R , b) and (b , R , a) hold, then it must be the case that (a b). This property has significant implications for the inclusion and exclusion of certain ordered pairs in the relation.

Step-by-Step Solution

Step 1: Analyze Given Pairs

(2, 2): Since this is a reflexive pair, it can always be included in the relation. (3, 4): This pair means that (4, 3) cannot be in the relation. (5, 6): Similarly, (6, 5) cannot be in the relation.

Step 2: Determine Remaining Pairs

For the set (A {1, 2, 3, 4, 5, 6}), after excluding the reflexive pairs (2, 2), we are left with the elements (1, 3, 4, 5, 6). We need to consider the reflexive and non-reflexive pairs in this subset.

Reflexive Pairs

(1, 1), (3, 3), (4, 4), (5, 5), (6, 6): Each of these pairs can be included or excluded independently. There are 5 such pairs, giving us (2^5 32) choices.

Non-Reflexive Pairs

For the non-reflexive pairs, we must ensure that the anti-symmetry condition is not violated. We need to exclude the pairs (3, 4) and (4, 3), and (5, 6) and (6, 5).

From (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (4, 1), (5, 1), (6, 1), we see that (4, 3) and (6, 5) cannot be included. From (3, 1), (4, 1), (5, 1), (6, 1), (1, 5), (3, 5), (4, 5), (6, 5), we see that (5, 3) and (5, 4) and (5, 6) cannot be included. From (4, 1), (5, 1), (6, 1), (1, 6), (4, 6), (5, 6), we see that (6, 4) and (6, 5) cannot be included. Finally, from (5, 1), (6, 1), (1, 6), (5, 6), we see that (6, 5) cannot be included.

This gives us a total of 12 valid non-reflexive pairs, each of which can be included or excluded, giving us (2^{12} 4096) choices.

Step 3: Total Number of Anti-Symmetric Relations

Combining the choices for reflexive and non-reflexive pairs, we have a total of (2^5 times 2^{12} 2^{17} 131072) anti-symmetric relations.

Final Answer

The number of anti-symmetric relations on the set (A {1, 2, 3, 4, 5, 6}) that contain the ordered pairs (2, 2), (3, 4), and (5, 6) is 131072.