How can we determine if two sets are equal?

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Understanding Set Equality

To say that two sets are equal, we must verify that they contain exactly the same elements. In mathematical terms, if A and B are two sets, they are considered equal, written as A = B, if every element of A is also an element of B, and vice versa.

Key Points to Consider

  • Order Doesn’t Matter: The arrangement of elements in a set does not affect its equality. For example, the sets {1, 2, 3} and {3, 2, 1} are equal since they contain the same elements.
  • No Duplicates: A set cannot have duplicate elements. Therefore, {1, 1, 2} is essentially the same as {1, 2}. If two sets contain duplicates, they are still considered equal as long as their unique elements match.
  • Subset Check: To confirm that two sets are equal, one effective method is to check if each set is a subset of the other.
    • For A to be a subset of B, each element in A must also be in B.
    • Similarly, for B to be a subset of A, each element in B must also be in A.

Examples

Let’s say we have two sets:

  • A = {2, 4, 6}
  • B = {4, 6, 2}

Both sets contain the same elements but in a different order, making them equal.

Now consider:

  • A = {1, 2, 3}
  • B = {2, 3, 4}

Here, the two sets are not equal since their elements differ.

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