Write the following statement in symbolic form, and then give (in symbolic form and in English) its contrapositive and its negation:
“If all birds can swim or some fish can fly, then no whales can walk”
<1>2. The original statement is \[ \left[(\forall x\,(B(x)\Rightarrow S(x)))\lor(\exists x\,(F(x)\land Y(x)))\right] \Rightarrow \left[\forall x\,(W(x)\Rightarrow \neg K(x))\right]. \]
<1>3. Its contrapositive is \[ \left[\exists x\,(W(x)\land K(x))\right] \Rightarrow \left[(\exists x\,(B(x)\land\neg S(x)))\land (\forall x\,(F(x)\Rightarrow\neg Y(x)))\right]. \]
<1>4. In English, the contrapositive is: “If some whale can walk, then some bird cannot swim and no fish can fly.”
<1>5. The negation of the original statement is \[ \left[(\forall x\,(B(x)\Rightarrow S(x)))\lor(\exists x\,(F(x)\land Y(x)))\right] \land \left[\exists x\,(W(x)\land K(x))\right]. \]
<1>6. In English, the negation is: “All birds can swim or some fish can fly, and some whale can walk.”
Solution
1. Let the universe be all animals, and let\[ B(x),S(x),F(x),Y(x),W(x),K(x) \]mean respectively that \(x\) is a bird, can swim, is a fish, can fly, is a whale, and can walk.
Write the original implication as \((A\lor C)\Rightarrow D\). Its contrapositive is \[ \neg D\Rightarrow\neg(A\lor C), \] and De Morgan’s law gives \(\neg(A\lor C)=\neg A\land\neg C\). Moreover, \[ \neg\forall x(B(x)\Rightarrow S(x)) \equiv \exists x(B(x)\land\neg S(x)), \] \[ \neg\exists x(F(x)\land Y(x)) \equiv \forall x(F(x)\Rightarrow\neg Y(x)), \] and \[ \neg\forall x(W(x)\Rightarrow\neg K(x)) \equiv \exists x(W(x)\land K(x)). \]
The negation of an implication \(P\Rightarrow Q\) is \(P\land\neg Q\). Apply this with the antecedent and consequent from 2, and use the final equivalence in 3.