Math Problem Statement

given A ={ a:a isba prime number<23 } then number of irreflexive relation onA is

Solution

The set AA consists of prime numbers less than 23. Let's first identify the prime numbers less than 23:

A={2,3,5,7,11,13,17,19}A = \{2, 3, 5, 7, 11, 13, 17, 19\}

So, AA has 8 elements.

Irreflexive Relations:

An irreflexive relation on a set AA means that no element in AA is related to itself. For each pair (a,a)(a, a) in AA, the relation must not include (a,a)(a, a).

The total number of ordered pairs that can be formed from the set AA is A2|A|^2, which is 82=648^2 = 64. However, since irreflexive relations exclude pairs of the form (a,a)(a, a), we have to exclude the 8 reflexive pairs (a,a)(a, a).

Thus, the number of ordered pairs that can be considered in an irreflexive relation is 648=5664 - 8 = 56.

Number of Irreflexive Relations:

For each of the 56 remaining pairs, we have two choices: either include the pair in the relation or not. Therefore, the number of possible irreflexive relations is:

2562^{56}

This is the total number of irreflexive relations that can be defined on the set AA.

Final Answer:

The number of irreflexive relations on the set AA is 2562^{56}.


Would you like further details or clarifications? Here are some related questions:

  1. What are reflexive relations and how do they differ from irreflexive relations?
  2. How can we define a symmetric relation on a set?
  3. What is the total number of relations on a set with nn elements?
  4. How do antisymmetric relations differ from symmetric ones?
  5. Can a relation be both reflexive and irreflexive? Why or why not?

Tip: When working with relations, always check if self-relations (like (a,a)(a, a)) are allowed or excluded depending on the type of relation being asked.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Relations
Prime Numbers

Formulas

Total number of ordered pairs: |A|^2
Number of irreflexive relations: 2^(|A|^2 - |A|)

Theorems

Irreflexive Relation Theorem

Suitable Grade Level

Grades 10-12