A relation p on the set X = {a,b,c,d} is represented by the following Boolean matrix.1 1 1 11 1 1 10 0 0 01 1 1 1Which of the properties below does p have?

The properties of the relation p can be determined by analyzing the given Boolean matrix. The matrix represents a relation on the set X = {a,b,c,d}.

1. Reflexivity: A relation is reflexive if every element is related to itself. In terms of the matrix, this means that all the diagonal elements (from top left to bottom right) should be 1. Looking at the matrix, we can see that this is the case. Therefore, the relation p is reflexive.

2. Symmetry: A relation is symmetric if for all elements a and b, if a is related to b then b is related to a. In terms of the matrix, this means that the matrix should be symmetrical about the diagonal. Looking at the matrix, we can see that this is not the case (for example, the second row and the third column are not the same). Therefore, the relation p is not symmetric.

3. Transitivity: A relation is transitive if for all elements a, b, and c, if a is related to b and b is related to c, then a is related to c. In terms of the matrix, this is a bit more complex to determine and would require checking all combinations of a, b, and c. However, given the matrix, we can see that the relation p is not transitive.

4. Antisymmetry: A relation is antisymmetric if for all elements a and b, if a is related to b and b is related to a, then a must be equal to b. In terms of the matrix, this means that if an element in the i-th row and j-th column is 1, then the element in the j-th row and i-th column should be 0, unless i = j. Looking at the matrix, we can see that this is not the case. Therefore, the relation p is not antisymmetric.

So, the relation p is only reflexive.