Math Problem Statement

Give an example of relation on {1,2,3,4} that is not reflexive, symmetric, not antisymmetric, and transitive.

Solution

To construct a relation on the set {1,2,3,4}\{1, 2, 3, 4\} that is not reflexive, not symmetric, not antisymmetric, and not transitive, we need to define the relation carefully to violate all of these properties.

Let's define the relation RR on the set {1,2,3,4}\{1, 2, 3, 4\} as follows:

R={(1,2),(2,3),(3,4),(4,1),(2,1)}R = \{(1, 2), (2, 3), (3, 4), (4, 1), (2, 1)\}

Now, let's check the properties:

  1. Not Reflexive: A relation is reflexive if for every element x{1,2,3,4}x \in \{1, 2, 3, 4\}, (x,x)(x, x) is in the relation. However, in this case, there are no pairs of the form (1,1)(1,1), (2,2)(2,2), (3,3)(3,3), or (4,4)(4,4) in RR. Hence, the relation is not reflexive.

  2. Not Symmetric: A relation is symmetric if whenever (a,b)(a, b) is in the relation, (b,a)(b, a) must also be in the relation. In RR, we have (1,2)(1, 2) but (2,1)(2, 1) is in the relation, which makes the relation appear symmetric. However, the pair (2,3)(2, 3) is in the relation, but (3,2)(3, 2) is not. So, the relation is not symmetric.

  3. Not Antisymmetric: A relation is antisymmetric if for any pair of elements (a,b)(a, b) and (b,a)(b, a) in the relation, a=ba = b. In our relation, we have both (1,2)(1, 2) and (2,1)(2, 1) in RR, but 121 \neq 2. Thus, the relation is not antisymmetric.

  4. Not Transitive: A relation is transitive if whenever (a,b)(a, b) and (b,c)(b, c) are in the relation, (a,c)(a, c) must also be in the relation. In this case, (1,2)(1, 2) and (2,3)(2, 3) are in RR, but (1,3)(1, 3) is not. Hence, the relation is not transitive.

Summary:

The relation R={(1,2),(2,3),(3,4),(4,1),(2,1)}R = \{(1, 2), (2, 3), (3, 4), (4, 1), (2, 1)\} on the set {1,2,3,4}\{1, 2, 3, 4\} is not reflexive, not symmetric, not antisymmetric, and not transitive.


Would you like more details on any of these properties or further examples?

Here are some follow-up questions to explore:

  1. What are the conditions for a relation to be reflexive, symmetric, antisymmetric, and transitive?
  2. Can you provide an example of a relation that is both symmetric and transitive but not reflexive?
  3. How would you construct a relation that is reflexive but neither symmetric nor transitive?
  4. What are the differences between antisymmetry and symmetry in relations?
  5. How do you check if a relation on a set is transitive?

Tip: To easily check for transitivity, try finding "chains" in the relation where (a,b)(a, b) and (b,c)(b, c) are present, and then verify whether (a,c)(a, c) is also in the relation.