This problem is in two parts. The
first uses logic in the context of English language whereas the
second uses logic in the clearer context of
Which of the
following are certainly true, which are certainly false. How many
statements form equivalent pairs? Are there any parts of the
problem which are debatable or unclear?
If my team wins the world cup tomorrow then I'll be happy
If I am happy tomorrow then my team will win the world cup
If I am not happy tomorrow then my team will not win the world
If my team does not win the world cup tomorrow then I will not
be happy tomorrow.
If this is maize then it grew from a seed
If this grew from a seed then it is maize
If this did not grow from a seed then it is not maize
If this is not maize then it did not grow from a seed
If Rover is a dog then Rover is an animal
If Rover is not an animal then Rover is not a dog
If Rover is not a dog then Rover is an animal
If Rover is an animal then Rover is a dog
These ideas will help you to understand part 2.
logic the implication arrows $\Rightarrow$ and $\Leftrightarrow$
are used to connect expressions as follows:
$p\Rightarrow q$ means 'IF $p$ is true THEN $q$ is true.
$p\Leftrightarrow q$ means both $p\Rightarrow q$ AND $q \Rightarrow
Convince yourself that
\left(p\Rightarrow q\right) \Leftrightarrow \left((NOT q)
\Rightarrow (NOT p)\right)
The expression on the right is called the contrapositive
of the statement on the
left. Since they are linked by $\Leftrightarrow$, proving one side
will automatically prove the other.
Consider these statements involving positive integers $n$ and
1: $n+m$ is odd $\Rightarrow n\neq m$.
2: $n+m$ is even $\Rightarrow$ $n$ and $m$ are either both even or
3: $n^2$ is even $\Rightarrow n$ is even.
4: $n^3$ is odd $\Rightarrow n$ is odd.
5: $n$ mod (4) = 2 or 3 $\Rightarrow$ $n$ is not a perfect
Write out the contrapositive versions of these statements and use
these to prove the statements. You can assume that an even number
can be written as $2N$ and an odd number as $2M+1$.