Math Problem Statement
given A ={ a:a isba prime number<23 } then number of irreflexive relation onA is
Solution
The set consists of prime numbers less than 23. Let's first identify the prime numbers less than 23:
So, has 8 elements.
Irreflexive Relations:
An irreflexive relation on a set means that no element in is related to itself. For each pair in , the relation must not include .
The total number of ordered pairs that can be formed from the set is , which is . However, since irreflexive relations exclude pairs of the form , we have to exclude the 8 reflexive pairs .
Thus, the number of ordered pairs that can be considered in an irreflexive relation is .
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:
This is the total number of irreflexive relations that can be defined on the set .
Final Answer:
The number of irreflexive relations on the set is .
Would you like further details or clarifications? Here are some related questions:
- What are reflexive relations and how do they differ from irreflexive relations?
- How can we define a symmetric relation on a set?
- What is the total number of relations on a set with elements?
- How do antisymmetric relations differ from symmetric ones?
- Can a relation be both reflexive and irreflexive? Why or why not?
Tip: When working with relations, always check if self-relations (like ) 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
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