GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 3

Question

Let's consider a propositional language where

• $p$ means "$x$ is a power of $2$",
• $q$ means "$x$ is even".

Which of the following is the correct formal representation of the sentence
"$x$ being a power of $2$ is a sufficient condition for $x$ being even"?

• A. $p \rightarrow q$
• B. $q \rightarrow p$
• C. $\neg p \rightarrow \neg q$
• D. none of these

Answer 1

The prepositional formula $p \rightarrow q$ represent the fact that if $p$ is true, so is $q$ or in other words $p$ is a sufficient condition for $q.$ $p$ is not a necessary condition for $q$ as $q$ can be true irrespective of the value of $p.$

Answer 2

• $p$ means "$x$ is a power of $2$",
• $q$ means "$x$ is even".

"$x$ being a power of $2$ is a sufficient condition for $x$ being even"

$\text{The following ways to express the conditional statement:} (p\rightarrow q)$

• “if $p$, then $q$”
• “$p$ implies $q$”
• “if $p, q$” 
• “$p$ only if $q$”
• “$p$ is sufficient for $q$” 
• “a sufficient condition for $q$ is $p$”
• “$q$ if $p$” 
• "$q$ whenever $p$”
• “$q$ when $p$” 
• “$q$ is necessary for $p$”
• “a necessary condition for $p$ is $q$”
• “$q$ follows from $p$”
• “$q$ unless $\neg p$” 
• “$q$ provided that $p$”

Now, $p$ is sufficient for $q \equiv p\rightarrow q$

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

Answer 3

p is sufficient condition for q is represented as:

p → q