Matrix relation of sets. symmetric, antisymmetric,reflexive,transitive

Since it contains (c,b) and (b,a) but not (c,a) it is not transitive.In summary, the relation A = {a,b,c} for the following matrix [1,0,0;1,1,0;0,1,1] is reflexive and antisymmetric, but not symmetric or transitive.
  • #1
sapiental
118
0

Homework Statement



relation A = {a,b,c} for the following matrix [1,0,0;1,1,0;0,1,1]

is it reflexive, transitive, symmetric, antisymmetric


Homework Equations



ordered pairs.

The Attempt at a Solution



i wrote the ordered pairs as (a,a),(b,a),(b,b),(c,b),(c,c)

I only that it is reflexive for a,a b,b and c,c
also it is antisymmetric because there are no edges in opposite directions between distinct verticies.

am I missing anything. thanks!
 
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  • #2
sapiental said:

Homework Statement



relation A = {a,b,c} for the following matrix [1,0,0;1,1,0;0,1,1]

is it reflexive, transitive, symmetric, antisymmetric


Homework Equations



ordered pairs.

The Attempt at a Solution



i wrote the ordered pairs as (a,a),(b,a),(b,b),(c,b),(c,c)

I only that it is reflexive for a,a b,b and c,c
also it is antisymmetric because there are no edges in opposite directions between distinct verticies.

am I missing anything. thanks!
I don't know what you mean by "reflexive for a,a b,b and c,c. A relation is reflexive if and only if it contains (x,x) for all x in the base set. Since only a, b, and c are in the base set, and the relation contains (a,a), (b,b), and (c,c), yes, it is reflexive.

To be symmetric, since it contains (b,a) it would have to contain (a,b) and it doesn't: not symmetric. Since it does NOT contain (a,b) or (b,c), yes, it is anti-symmetric.

What about transitive? A relation is transitive if and only if, whenever (x,y) and (y,z) are in the relation, so is (x,z). Can you find pairs so that is NOT true?
 
  • #3
Hey, thanks for the reply!

I didn't put parenthesis around the ordered pairs (a,a),(b,b),(c,c) for the first problem, sorry.

I don't think it's transitive since we have (c,b) and (b,a), and it doesn't contain (c,a). How does that sound? Thanks
 
  • #4
Yes, that completes it.
 

1. What is the matrix representation of sets?

The matrix representation of sets is a method of representing relationships between elements of a set using a matrix, where each row and column represents an element of the set and the values in the matrix indicate whether there is a relationship between those elements.

2. What is a symmetric relation in a matrix representation?

A symmetric relation in a matrix representation is one where the values in the matrix are the same when reflected across the main diagonal. This means that for any two elements in a set, if there is a relationship between them, the relationship is bidirectional.

3. What is an antisymmetric relation in a matrix representation?

An antisymmetric relation in a matrix representation is one where the values in the matrix are opposite when reflected across the main diagonal. This means that for any two elements in a set, if there is a relationship between them, the relationship is only one-directional.

4. What is a reflexive relation in a matrix representation?

A reflexive relation in a matrix representation is one where the values on the main diagonal are all 1's. This means that every element in a set is related to itself.

5. What is a transitive relation in a matrix representation?

A transitive relation in a matrix representation is one where the values in the matrix follow the transitive property. This means that if element A is related to element B and element B is related to element C, then element A is also related to element C.

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