Math Problem Statement

explain difference b/w symmetric, asymmetric and antisymmetric sets in simple language with simplest set example

Solution

Let's break down the concepts of symmetric, asymmetric, and antisymmetric sets with simple explanations and examples.

1. Symmetric Relation:

  • Definition: A relation RR on a set AA is symmetric if, whenever an element aa is related to an element bb (i.e., aRbaRb), then bb is also related to aa (i.e., bRabRa).
  • Example:
    • Set A={1,2}A = \{1, 2\}
    • Relation R={(1,2),(2,1)}R = \{(1, 2), (2, 1)\}
    • Here, since (1,2)R(1, 2) \in R and (2,1)R(2, 1) \in R, the relation is symmetric.

2. Asymmetric Relation:

  • Definition: A relation RR on a set AA is asymmetric if, whenever an element aa is related to an element bb (i.e., aRbaRb), then bb is not related to aa (i.e., bRabRa does not hold). Asymmetric relations are not symmetric at all.
  • Example:
    • Set A={1,2}A = \{1, 2\}
    • Relation R={(1,2)}R = \{(1, 2)\}
    • Here, (1,2)R(1, 2) \in R but (2,1)R(2, 1) \notin R, so the relation is asymmetric.

3. Antisymmetric Relation:

  • Definition: A relation RR on a set AA is antisymmetric if, whenever both aRbaRb and bRabRa hold, then aa must be equal to bb. This means if aa is related to bb and bb is related to aa, they must be the same element.
  • Example:
    • Set A={1,2}A = \{1, 2\}
    • Relation R={(1,1),(2,2),(1,2)}R = \{(1, 1), (2, 2), (1, 2)\}
    • Here, (1,2)R(1, 2) \in R but (2,1)R(2, 1) \notin R. Also, (1,1)R(1, 1) \in R and (2,2)R(2, 2) \in R, so the relation is antisymmetric.

Summary:

  • Symmetric: If aa is related to bb, then bb is related to aa.
  • Asymmetric: If aa is related to bb, then bb is not related to aa.
  • Antisymmetric: If aa is related to bb and bb is related to aa, then aa must be equal to bb.

Would you like more details or have any questions?

Related Questions:

  1. Can a relation be both symmetric and antisymmetric?
  2. What is a real-life example of a symmetric relation?
  3. How does transitivity relate to symmetric and antisymmetric relations?
  4. What is a reflexive relation, and how does it differ from symmetric relations?
  5. Can a relation be both asymmetric and antisymmetric?

Tip:

When identifying the type of relation, always check pairs carefully to see if the condition holds for all elements in the set.