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Definition 1: If two sets A and B have the same cardinality if there exists an objective function from set A to B. Definition 2: Two sets A and B are said to be equivalent if they have the same cardinality i.e. n(A) = n(B). In general, we can say, two sets are equivalent to each other if the number of elements in both the sets is equal.
12 Ιουλ 2024 · Equivalent Sets. Two or more sets are said to be equivalent if they have the same number of elements, regardless of what the elements are. Thus, two equivalent sets have the same cardinality, which means the elements of both sets correspond to each other on a one-to-one basis.
21 Νοε 2023 · Equivalent sets are sets that contain the same number of elements, although the elements themselves may be different. For example, set A {5, 10, 15, 20} is equivalent to set B {w,...
7 Σεπ 2021 · A set \ (A\) is a subset of \ (B\text {,}\) written \ (A \subset B\) or \ (B \supset A\text {,}\) if every element of \ (A\) is also an element of \ (B\text {.}\) For example, \ [ \ {4,5,8\} \subset \ {2, 3, 4, 5, 6, 7, 8, 9 \} \nonumber \] and.
An equivalent set does not require the elements to be identical; it is solely about having the same number of members. The symbol for equivalent sets is usually denoted by `\sim` or `\approx`, indicating that the sets being compared have equal cardinality.
But, a nice example of geometric projection — vertical projection (a.k.a. \(π_1\)) — can be used to show that (for example) the interval \((−1, 1)\) and the portion of the unit circle lying in the upper half-plane are equinumerous.
Using the following theorem, we can combine certain sets that we know are equivalent to conclude that other sets are also equivalent (without needing to explicitly produce a bijection between them).
