# Discrete math: Inverse, converse, contrapositive - simplifying expressions

State the inverse, converse, and contrapositive of the following implication expression as English sentences. Ensure that you list the symbols you will use for each ATOMIC predicate. You must also simplify your expressions and distribute your negation operations as much as possible before translating.

Given statement: If it is entertaining then it is not impossible and it is not difficult

So I get how to do the inverse, converse, and contrapositive, but I don't really know what my practice worksheet means by "simplify your expressions and distribute your negation operations as much as possible before translating."

Like if I were to find the contrapositive I would do:

Let e be the predicate "if it is entertaining", i be "it is not impossible" and d be "it is not difficult"

$$e → (i\lor d)$$

$$¬(i \lor d) → ¬e$$

$$(¬i \land ¬d) → ¬e$$

Translation: If it is impossible and difficult then it is not entertaining.

but I don't know if that meets the requirements for "simplifying the expression and distribute the negation operations as much as possible before translating."

• I think distribute the negation operations refers to changing $\lnot (i \lor d)$ to $\lnot i \land \lnot d$, as you did – J. W. Tanner May 16 at 2:09

Also I'd use $$i$$ and $$d$$ for "it is impossible" and "it is difficult", and use $$\lnot$$ to handle the "not" part.
This statement is: $$~e\to(\lnot i~\land~\lnot d)$$