How to Express Sets with Specific Cardinality Restrictions?

[xeon123]

In the expression of sets: B={X $\in$ A:|X|<3} the expression is saying that B is a set that contains at most 3 sets X that belongs to A, right?

How do we say, B is a set that contains elements of X that belongs to A, and all X elements contains at most 3 x elements (the cardinality of X is at most 2). For example, B={{1,2},{3}}. And not, B={{1,2,3}}

 

[tiny-tim]

hi xeon123!

In the expression of sets: B={X $\in$ A:|X|<3} the expression is saying that B is a set that contains at most 3 sets X that belongs to A, right?

no, it's saying that B is the collection of all elements of A with less than 3 elements

 

[haruspex]

How do we say, B is a set that contains elements of X that belongs to A, and all X elements contains at most 3 x elements (the cardinality of X is at most 2). For example, B={{1,2},{3}}. And not, B={{1,2,3}}

Assuming you meant "B is the set that consists of elements X that belong to A and which contain at most 2 elements" then: B={X $\in$ A:|X|<3}
Your example works provided that the only elements of A with less than three elements are the two you show.