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Transcript
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}.