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'Contrary Logic' printed from https://nrich.maths.org/
We got two great solutions from Alex, from
Stoke-on-Trent Sixth Form College and Jamie from the Omagh Academy.
The solution here is a combination of their responses
To begin, Alex formally split the worded
statements into their correct logical expressions and correctly
noted that several pairs were logically equivalent
Let p represent my team winning the world cup tomorrow, and q
represent my happiness tomorrow. The statements can be written in
terms of p and q as follows:
p => q
q => p
(NOT q) => (NOT p)
(NOT p) => (NOT q)
The first and third statements are logically identical, as are the
second and fourth, by the logical rule in part 2 of the question.
If p represented the object being maize, and q represent it having
grown from a seed then the same relationship between the statements
would apply: they are again logically identical.
If p represents that Rover is a dog, and q represents that Rover is
an animal then the statements can be written in terms of p and q as
follows:
p => q
(NOT q) => (NOT p)
(NOT p) => q
q => p
The first and second statements are logically equivalent identical,
the same as with the world cup and maize parts.
Jamie gave us an excellent discussion on
whether the statements were, or were not, true or false. Jamie
quite rightly noted that the statements were not really clear
enough to be declared True or False and, rather nicely, suggested
ways in which the statements could be altered to make more
sense
If my team wins the world cup
tomorrow then I'll be happy tomorrow.
This statement is unclear or debatable, since although his team
winning the world cup tomorrow may make him happy, something else
could happen which may make him unhappy: some event may befall him.
It is unsound because he does not know what tomorrow will
bring.
If my team does not win the world
cup tomorrow then I will not be happy tomorrow.
This statement is the contrapositive argument of the above
statement, and is also debatable for the same reason, that
something may happen tomorrow that may make him happy, yet he
assumes he will be entirely sad if his team loses.
If I am happy tomorrow then my
team will win the world cup tomorrow
Ths statement is false because it is saying that the man's
happiness tomorrow is a necessary and sufficient condition for his
team to win the world cup, despite the fact that his happiness has
no plausible effect on the outcome of the match.
If I am not happy tomorrow then my
team will not win the world cup tomorrow
Again this statement is false as it is saying that the man's
happiness tomorrow is a necessary and sufficient condition for his
team to win the world cup. It forms a pair with the previous
statement.
If this is maize then it grew from
a seed
This is true since maize must grow from a seed it is a plant, and
all plants grow from a seed. Otherwise, if it did not grow from a
seed, it cannot be maize, so the next statement (if this did not grow from a seed then it is
not maize ) must also be true, and so they form an
equivalent pair.
If this grew from a seed then it
is maize
This is not true: although maize grows from a seed, it is possible
that it could be any other plant that also grows from a seed. A
true statement would be 'If this grew from a seed, then it could be
maize.'
If this is not maize then it did
not grow from a seed
This is false, since it could be any other plant that does grow
from a seed.
If Rover is a dog then Rover is an
animal
This statement is true since all dogs are animals.
If Rover is not an animal then
Rover is not a dog
This statement is also true since all dogs are animals, so if Rover
is not an animal, Rover cannot be a dog. This statement goes along
with the previous statement, since they both work on the fact that
all dogs are animals.
If Rover is not a dog then Rover
is an animal
This statement is false: saying that Rover is not a dog is
equivalent to saying that Rover could be anything apart from a dog;
since not everything is an animal, the statement is
false.(Editor: I love this
answer!)
If Rover is an animal then Rover
is a dog
This is false since clearly not all animals are dogs. A true
statement would be 'If Rover is an animal, then Rover could be a
dog'.
It has been noted that it might be possible
to argue very pedantically against all of these statements. This is
because they are phrased in English language with all of its
vagueness and imprecision. The world cup statement clearly only
makes sense 'conversationally' and brings
to bear ideas that in usual language statements are neither
entirely true nor entirely false. Even the second and third sets of
statements are subject to some debate: Although several of the
staments might be very likely to be true or false, perhaps they
might be false for the following, albeit improbable reasons
:
What if the maize were grown in a lab as a
clone?
What if maize were a girl's
name?
What if Rover is the name of a play, and
the word 'dog' used in the slang sense to refer to something
worthless or of extremely poor quality?
Jamie and Alex both went on to give great
answers to the mathematically sound second part, demonstrating the
clarity of sound logical thinking so important to
mathematicians
1. $ (n+m)\mbox{ odd}\Rightarrow n\neq m$
The negation of $n+m$ odd is $n+m$ even and the negation of $n\neq
m$ is $n=m$, so the first contrapositive statement is
$$ n=m\Rightarrow (n+m)\mbox{ even}$$
To prove this, note that when $n=m$, $n+m$ is equivalent to $2n$,
which is even. This proves the contrapositive statement, so the
initial statement must also be true.
2.$(n+m)\mbox{ even}\Rightarrow n \mbox{ and } m \mbox{ are either
both even or both odd}$
The negation of ($n$ and $m$ are either both even or both odd) is
(one of $n$ and $m$ is even, and the other odd), so the
contrapositive statement is
$$(\mbox{one of } n \mbox{ and } m \mbox{ is even, and the other
odd})\Rightarrow (n+m) \mbox{ is odd}$$
To prove this, write the even integer as $2N$, and the odd one as
$2M+1$, so that their sum is $2N+2M+1 = 2(N+M)+1$. This is odd, and
the statement is therefore proved.
3. $n^2$ is even $\Rightarrow n$ is even.
The contrapositive of this statement is ($n$ odd $\Rightarrow n^2$
odd)
To prove this assume that $n$ is odd so that $n = 2M+1$. Then $n^2
= (2M+1)^2 = 4M^2+4M+1$, which is odd. This proves the
result.
4. $n^3$ is odd $\Rightarrow n$ is odd.
The contrapositive of this statement is ($n$ even $\Rightarrow n^3$
even)
To prove this assume that $n$ is even so that $n=2N$. Then $n^3=8N$
which is even. So the result is proved.
5. $n \mbox{ mod }(4) = 2 \mbox{ or } 3 \Rightarrow n$ is not a
perfect square.
The contrapositive of this is
($n$ is a perfect square $\Rightarrow n \mbox{ mod }(4) = 0 \mbox{
or } 1$)
To prove this start by noting that since $n$ is a perfect square,
its square root is a whole number. So, $\sqrt{n}$ is either even or
odd, and so one of $2N$ or $2M+1$. The square of $2N$ is $4N^2$,
which is congruent to $0 \mbox{ mod }4$. The square of $2M+1$ is
$4(M^2 + M)+1$, which is congruent to $1 \mbox{ \mod }4$. Therefore
$n$ is congruent to $(0 \mbox{ mod }4)$ or $(1 \mbox{ mod }4)$.
This proves the result.