Question

Here are two new definitions. Two schemata are compatible just in case their conjunction is satisfiable....

Here are two new definitions.
Two schemata are compatible just in case their conjunction is satisfiable. Two schemata are congruent just in case their biconditional is satisfiable.

Determine whether each of the generalizations below is true or false. If it is true, explain why it is true. If it is false, present a counterexample.

(a) If two schemata X and Y are compatible, then they are congruent. (b) If two schemata are congruent, then they are compatible.

Science   Advanced-Physics   
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Answer #1

Ans. We will start with an example. If two triangles are congruent, then they are similar, because when two triangles are congruent, then all corresponding sides as well as corresponding angles of one triangle are equal to those of the other triangle. But if two triangles are similar, then they may or may not be congruent, because for triangles to be similar, we just need all angles to be equal, but for triangles to be congruent, all angles as well as all sides should be equal.Hence while congruent triangles are similar, similar triangles may or may not be congruent.

In the same sense, if two schemata are congruent, then they are compatible. But if two schemata are compatible, then they may or may not be congruent.

Hence statement (a) is false, whereas statement (b) is true.

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