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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.
Contrary Logic
Age 16 to 18
Challenge Level 
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.
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