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COMPARING SETS Definition: EQUALITY OF 2 SETS Two sets A and B are equal (written A = B) if they have exactly the same elements. Otherwise, we write A ≠ B. Example: {1, 2, 3, 7} = {1, 7, 3, 2} (Note that order is not important.) Example: {a, b, c} ≠ {a, c, e} Definition: SUBSET Set A is a subset of set B (written A B) if every element of A is also an element of B. Ex: {a, b, e} {a, b, c, e, g} Ex: {a, b, c, e, g} {a, b, c, e, g} Ex: {dog, cat} is NOT a subset of {dog, pig, goat} Definition: PROPER SUBSET Set A is a proper subset of set B (written A B) if every element of A is also an element of B but A ≠ B. Ex: {a, b, e} {a, b, c, e, g} Ex: {a, b, c, e} is NOT a proper subset of {a, b, c, e} Ex: {d, c} is NOT a proper subset of {d, p, g} COMPARING SETS Important: Pay close attention to the difference between subset (“ ”) and proper subset (“ ”). It might help to make an analogy to something with which you are more familiar: Difference between less than (“<”) and less than or equal to (“<”) Just as “<” allows the possibility of equality: (5 < 5 is a true statement) “ ” also allows for the possibility of equality: ({1, 4, 7} {1, 4, 7} is a true statement) A proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality. Number of subsets of a set: A set with k elements has subsets. Ex: For the set {a, c, f}, there are =8 elements. Let’s list them: Note: The empty set & A itself are both subsets of any set A so they both are included in the subset count. Subsets with: 0 elements 1 element 2 elements 3 elements Ø {a}, {c}, {f} {a,c},{a,f},{c,f} {a, c, f} Definition: EQUIVALENCE OF 2 SETS Two sets A and B are equivalent if the two sets have the same number of elements…that is, if n(A) = n(B). Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels. Some analog clock use ordinary numbers while some use Roman numerals: A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII} It is in this sense that we say that set A is equivalent to set B. Ex: {a, 2, 5, f} is equivalent to {dog, cat, bird, pig}. Example: {p, q, x} is NOT equivalent to {a, 2, 5, f}.