Discrete math contrapositive statment of universal statement

Question

I have researched many sides about it but could not find exact answer. My professor asked me to write contrapositive statment and convert statment of statment “ all red cars are fast “ Meanwhile he only explained how to do that base on conditional statment which has hypothesis and conclusion. My question is how to solve this example ? Is it possible to make universal or even existential statment to contrapositive or converse or even inverse stAtment ? If yes than how ?

Answer 1

We can tranlate "All cars are fast" to "If x is a red car, then x is fast."

Domain of $x$ "all things". $C(x):$ x is a car. $F(x)$: x is fast. $R(x):$ "x is red."

Hence, we have $$\forall x\Big( (C(x)\land R(x)) \to F(x)\Big)\tag 1$$

Thus the contrapositive of $(1)$, which is equivalent to $(1)$ becomes: $$\forall x\big(\lnot F(x) \to \lnot (C(x)\land R(x))\big)\tag 2$$ (using the equivalence: $p\to q \equiv \lnot q \to \lnot p$). We can read this as "Anything that's not fast is not a red car."

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Note that $(2)$ can also be written as follows: $$\begin{align}\forall x\Big(\lnot F(x) \to \lnot (C(x)\land R(x))\Big)&\equiv \forall x\Big(F(x) \lor \lnot\big(C(x) \land R(x)\big)\Big) \\ \\ &\equiv \forall x \lnot\Big(\lnot F(x)\land (C(x) \land R(x))\Big)\\ \\ &\equiv \lnot \exists x \Big((C(x) \land R(x)) \land \lnot F(x)\Big)\end{align}$$

The last expression would read "No red car is not fast."

Answer 2

Let's set $R$ to be the set of red cars, and $F$ all the things that are fast. So, this statement would translate to $$x\in R\to x\in F$$ The contrapositive is found by inversing the conditional and negating it, which makes $$x\notin F\to x\notin R$$Which can be translated to "If it isn't fast, it isn't a red car"