GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 26

Question

For every athlete $A,$ if $A$ is successful then $A$ has good skills and $A$ is hardworking.
Which one of the following sentences is equivalent to the above claim.

• A. For every athlete $A$, if $A$ does not have good skills or $A$ is not hardworking, then $A$ is not successful
• B. There is an athlete $A,$ such that if $A$ does not have good skills or $A$ is not hardworking, then $A$ is not successful
• C. If $A$ does not have good skills or $A$ is not hardworking, then $A$ is not successful
• D. None of these

Answer 1

For every athlete $x,$ if $x$ is successful then $x$ has good skills and $x$ is hardworking.

Lets, $S(x):x$ is successful$,G(x):x$ has good skill $,H(x):x$ is hardworking.

We can write , $\forall x \left[S(x) \rightarrow (G(x) \wedge H(x))\right] \equiv \forall x[\neg S(x) \vee (G(x) \wedge H(x))]$

Option $A.$ For every athlete $x$, if $x$ does not have good skills or $x$ is not hardworking, then $x$ is not successful.

We can write, $\forall x \left[(\neg G(x) \vee \neg H(x)) \rightarrow \neg S(x) \right] \equiv \forall x[\neg (\neg G(x) \vee \neg H(x)) \vee \neg S(x)]\equiv \forall x[\neg S(x) \vee (G(x) \wedge H(x))]$

Option $B.$ There is an athlete $x,$ such that if $x$ does not have good skills or $x$ is not hardworking, then $x$ is not successful.

We can write, $\exists x[(\neg G(x) \vee \neg H(x)) \rightarrow \neg S(x)]\equiv \exists x[\neg (\neg G(x) \vee \neg H(x)) \vee \neg S(x)]\equiv \exists x[\neg S(x) \vee (G(x) \wedge H(x))]$

Option $C.$ If $x$ does not have good skills or $x$ is not hardworking, then $x$ is not successful.

$\left[(\neg G(x) \vee \neg H(x)) \rightarrow \neg S(x) \right] \equiv  x[\neg (\neg G(x) \vee \neg H(x)) \vee \neg S(x)]\equiv  x[\neg S(x) \vee (G(x) \wedge H(x))]$

So, the correct answer is $(A).$

Answer 2

A is the contrapositive of the given sentence and hence they are equivalent.