The set {1,2,3,4,5} is equivalent to

Two sets are said to be equivalent if they have exactly the same elements. The order of the elements in a set does not matter, and repeating an element in set notation does not introduce a new element. What matters is which unique items are present.

Consider the set \(\{1,2,3,4,5\}\). An equivalent set must have each of these numbers, and no others, represented exactly once (since duplicates are ignored in set theory), and cannot be missing any elements.

• The set \(\{2,3,1,4\}\) is not equivalent because it is missing the number 5.
• The set \(\{1,2,3,4,5,5\}\) is equivalent to \(\{1,2,3,4,5\}\) because although it has an extra 5, duplicates are ignored in sets. Both sets represent exactly the same collection: numbers 1 through 5.
• The set \(\{1,2,3,4\}\) is not equivalent because it does not include 5.
• The set \(\{4,3,1,5,2\}\) is equivalent because it contains exactly the numbers 1, 2, 3, 4, and 5 (just in a different order, which does not matter in sets).

The key idea is: For two sets to be equivalent, their elements must match exactly (ignoring order and duplicates). So, \(\{4,3,1,5,2\}\) and \(\{1,2,3,4,5,5\}\) are both equivalent to \(\{1,2,3,4,5\}\), but any set missing an element or with a different element is not equivalent.