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Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
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In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element x in X is the subset of all elements in X which are equivalent to x. Equivalence classes among elements of a structure are often used to produce a smaller structure whose elements are the classes, distilling a relationship every element of the class shares with at least one other element of another class. This is known as modding out by the class. The class may assume the identity of one of the original elements, as when fractions are put in reduced form.
An equivalence relation is a binary relation ~ satisfying three properties:
The equivalence class of an element a is denoted [a] and may be defined as the set
of elements that are related to a by ~. The alternative notation [a]R can be used to denote the equivalence class of the element a specifically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.
The set of all equivalence classes in X given an equivalence relation ~ is denoted as X/~ and called the quotient set of X by ~. Each equivalence relation has a canonical projection map, the surjective function π from X to X/~ given by π(x) = [x].
This operation can be thought of (very informally) as the act of dividing the input set by the equivalence relation, hence both the name "quotient", and the notation, which are both reminiscent of division. One way in which the quotient set resembles division is that if X is finite and the equivalence classes are all equinumerous, then the number of equivalence classes in X/~ can be calculated by dividing the number of elements in X by the number of elements in each equivalence class. The quotient set X/~ may be thought of as the set X with all the equivalent points identified.
Every element x of X is a member of the equivalence class [x]. Every two equivalence classes [x] and [y] are either equal or disjoint. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongs to one and only one equivalence class. Conversely every partition of X comes from an equivalence relation in this way, according to which x ~ y if and only if x and y belong to the same set of the partition.
It follows from the properties of an equivalence relation that
In other words, if ~ is an equivalence relation on a set X, and x and y are two elements of X, then these statements are equivalent:
If ~ is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x ~ y, P(x) is true if P(y) is true, then the property P is said to be an invariant of ~, or well-defined under the relation ~.
A frequent particular case occurs when f is a function from X to another set Y; if f(x1) = f(x2) whenever x1 ~ x2, then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in the character theory of finite groups. Some authors use "compatible with ~" or just "respects ~" instead of "invariant under ~".
Any function f : X → Y itself defines an equivalence relation on X according to which x1 ~ x2 if and only if f(x1) = f(x2). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is the inverse image of f(x). This equivalence relation is known as the kernel of f.
More generally, a function may map equivalent arguments (under an equivalence relation ~X on X) to equivalent values (under an equivalence relation ~Y on Y). Such a function is known as a morphism from ~X to ~Y.