# For the relation below specify the properties (I.E. reflexive, symmetric, antisymmetric, transitive) they have Let A = { set of all people }, relation R: A x A where R = { (a,b) | a is at least...

For the relation below specify the properties (I.E. reflexive, symmetric, antisymmetric, transitive) they have

Let A = { set of all people }, relation R: A x A where R = { (a,b) | a is at least as tall as b }

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### 1 Answer

Given A={set of all people} with relation R:AxA and R={(a,b)|a is at least as tall as b}:

(1) A relation is reflexive R(a,a). (If the relation is denoted ~ then a~a).

Here a is at least as tall as a, so the relation is reflexive.

(2) A relation is symmetric if R(a,b) and R(b,a) for all a,b.

Here this would mean that if a is at least as tall as b, then b is at least as tall as a which is clearly not true. The relation is not symmetric.

(3) A relation is antisymmetric if R(a,b) and R(b,a) then a=b.

Here this means that if a is at least as tall as b, and b is at least as tall as a, then a and b are the same. Since the set is all people, the set is not antisymmetric as two different people could be the same height. (If the set were the heights of all people, the relation would be antisymmetric.)

(4) A relation is transitive if R(a,b) and R(b,c) implies R(a,c).

Here if a is at least as tall as b, and b is at least as tall as c, then a is at least as tall as c which is true.

The relation is transitive.

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