Games involving ``implies'' and ``equivalent''

The word ``implies'': We know that often if a claim is true, it is imperative that another claim, connected to the first claim, is also true. In this case we say that the first claim implies the second.

Examples: a) The claim ``The number formed from the last two digits of an integer is divisible by four'' implies the claim ``The given number itself is divisible by four''.

b) The claim ``Two angles are symmetric with respect to a given point in the plane'' implies the claim ``The two angles are equal''.

c) The claim ``The quadrilateral ABCD is a square'' implies the claim ``The quadrilateral ABCD is a rectangle.

In these and all similar cases it is enough to know that the first claim is true in order to be sure that the second claim is true. This is so because the truthfulness of the second claim is ``encapsulated'' somehow in the truthfulness of the first.

This allows us to formulate games, related to the word ``implies'', that test the understanding and logic reasoning of the kids. Common substitutes of the word ``implies'' are ``this means that...'', ``if... then...'', ``it follows that...'' etc.

Game 1. Each of four students is given a cardboard, on each of which one of the letters N, Z, D, and Q. This means that each student represents one of the sets. One of the participants, e. g. the one having the cardboard with N, is shown a cardboard with a number on its one side, e. g. 3, and x on its other side. If the student with the cardboard with N knows the natural numbers, he claims that the shown number belongs to him. The other three students, without having seen the number, must claim that it also belongs to them, since $x \in N$ implies $x \in Z$, $x \in D$, $x \in Q$.

If the student N ``declines'' the number x, it is shown to somebody else, e. g. D. If D accepts it, it follows that only Q must also accept it since $x \in D$ does tot imply that $x \in Z$, i. e. x can be fractional. If D does not accept the number either, it is shown to Z. If the latter accepts the number, Q must accept it as well.

The game with the number 3 can be repeated with other numbers like 1/3, -4, 2/5 etc. For each correct answer a point is given, for each wrong a point is deducted. The game can be repeated with other 4 students.

Game 2. It is analogous to the first. Instead of N, Z, D, and Q, we put on the cardboards the letters K - the set of all squares, D - the set of all diamonds, R - the set of all rectangles, and P - the set of all parallelograms. Cardboards on which a square, a diamond, a rectangle, or a parallelogram are shown.

If the teacher finds it appropriate, he or she may introduce the symbol $\Rightarrow$, explaining that it is often used as a substitute of the word ``implies''.

Equivalent claims. In mathematics as well as in everyday life we often use claims, meaning the same thing, e. g.

a) ``Every river has springs'' and ``There is no river that has no springs''.

b) ``There is a rectangle which is not a square'' and ``Not every rectangle is a square''.

Such pairs of claims we call equivalent. We use the symbol $\Leftrightarrow$ to denote the equivalence of the claims.

Using different equivalent claims, we can formulate different games.

Game 3. Ten students are given cardboards with the following claims on them:

a) Every natural number is an integer;
b) Every integer is a rational number;
c) Every natural number is a fraction;
d) There is at least one fraction which is not an integer;
e) Not every integer is a natural number;
f) There is no integer which is not a rational number;
g) Not every fraction is an integer;
h) There is at least one integer which is not a natural number;
i) There is no natural number which is not an integer;
j) There is no natural number which is not a fraction.

Then the students are asked to find the pairs of equivalent claims.

Other groups of equivalent claims can be used, e. g.:

Group A

a) There is at least one fraction which is not a natural number;
b) The sun is made of peanut butter;
c) Not every fraction is a natural number;
d) Every fraction is a natural number;
e) There is no natural number which is not a rational number;
f) there is at least one rational number which is not integer;
g) Every natural number is a rational number;
h) Not every rational number is an integer;
i) Not every rational number is a natural number;
j) There is at least one rational number which is not an integer.

Group B

a) Every square is a rectangle;
b) There is a diamond which is not a square;
c) There is no rectangle which is not a parallelogram;
d) Not every diamond is a square;
e) There is no square which is not a rectangle;
f) Every rectangle is a parallelogram;
g) Every diamond is a parallelogram;
h) Not every parallelogram is a diamond;
i) There is no diamond which is not a parallelogram;
j) There is a parallelogram, which is not a diamond.

To involve more students in the game, two different teams may be formed and given a set of equivalent claims. The winner is the team which successfully finds more pairs of equivalent claims for a restricted time.