High School What Does the Notation = in Equivalence Classes Conclude to?

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The notation "=" in equivalence classes indicates that two classes, [a] and [m], are identical, meaning they contain the same elements. The confusion arises from whether [a] should be a subset of [m] instead, but the equality signifies that every element in [a] is also in [m] and vice versa. The expression a ∈ [m] means that element a belongs to the equivalence class of m, implying that a is related to m under the equivalence relation. Using transitivity, if b is related to a (b ∼ a), then b is also related to m (b ∼ m), leading to the conclusion that all elements c related to m (c ∼ m) are also in the same equivalence class. Understanding these relationships clarifies the structure of equivalence classes in the context of the given notation.
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Please refer to the video at 37:02 from the link above.

I am struggling with the notation "=" of the property (a) which concludes to [a]=[m].
shouldn't it be [a]⊆[m], just like [m]⊆M.
 
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What does ##a \in [m]## mean? And what can be said about all elements ##b \sim a## if we use transitivity of ##\sim##?
In retrun, what does it mean for all elements ##c \sim m##?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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