In 9-33, determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answer.
Let A be the set of all English statements. A relation I is defined on A as follows: For every ,
To justify whether the given relation is reflexive, symmetric, transitive, or none of these.
Let A be the set of all English statements. A relation I Is defined on A as follows:
For all p, q ∈ A, p I q ⇔ p → q is true.
The relation I Is reflective if for every element
Let p be an English statement.
If p is true, then is true, since both (sub) proportions are true.
If p is false, then is true, since the if-statement is false.
Thus is always rue, which implies that and thus I Is reflexive.
The relation I on a set A is symmetric if
Let us assume that , by definition of I:
However, when p is false and q is true, then will be false and thus is not necessarily true, which implies that I Is not symmetric
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