It is not uncommon to hear the words "inverse" and "converse" used interchangeably, especially in their noun and adverb forms. You may have done this yourself. However, from a technical standpoint, they are not synonyms. The two words describe two very distinct types of relationships in logic, mathematics and statistics.
Inverse

In logical terms, the inverse of an object or statement is its opposite implication. If you state a condition of, "If X, then Y," the inverse of that conditional would be, "If not X, then not Y." For example, if you were to tell someone, "If it rains today, then I will stay inside and watch TV," the inverse of that statement would be to say, "If it does not rain today, then I will not stay inside and watch TV."
Converse

A conversion, on the other hand, is a type of logical contraposition describing an inference that can be derived from a categorical statement. For example, if you were to state that "No cats are dogs," it would conversely follow that "No dogs are cats." Similarly, if you were to say that "Some Greeks were philosophers," it would conversely follow that "Some philosophers were Greek." The second statement in each pairing can be automatically derived from the first.
The Logical Difference

The key difference between the two types of relationships is in their capacity for logical derivation. Unlike a converse relationship, you cannot derive the inverse of a conditional statement from the original statement. In saying, "If it rains today, then I will stay inside and watch TV," it does not automatically follow that "If it does not rain today, then I will not stay inside and watch TV." You may stay inside even if it does not rain.
Applications

In practice, an inverse relationship is used to denote a mathematical opposite. For example, the additive inverse of x would be x. The multiplicative inverse of x would be 1/x. A converse relationship is used in statistics to denote a correlation between two measured attributes. If the value of one variable you are measuring decreases in a consistent and statistically significant fashion as another variable increases, the two can be said to have a converse relationship.
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