Practice Empty Set in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The empty set, denoted or , is the unique set that contains no elements at all. It is a subset of every set because the statement 'every element of belongs to ' is vacuously true — there are no elements to contradict it.
Think of an empty box that is still a valid box—it just holds nothing. The empty set plays the same role for sets that zero plays for numbers: it is the identity element for union () and the annihilator for intersection (). It is also a subset of every set, which keeps logical statements about 'all elements of ' vacuously true.
Showing a random 20 of 50 problems.
Example 1
easyExample 2
easyIs a subset of ?
Example 3
mediumHow many elements are in ?
Example 4
mediumList all subsets of .
Example 5
mediumIf , what are and called, and can both be nonempty?
Example 6
mediumHow many subsets does the empty set have? List them.
Example 7
mediumIf , what is ?
Example 8
mediumSimplify .
Example 9
mediumIf , what must be?
Example 10
easyIs ?
Example 11
mediumTrue or false: 'every element of is a horse' is true.
Example 12
mediumWhat is the product of all elements in ?
Example 13
easyIs empty?
Example 14
mediumDistinguish the cardinalities: , , .
Example 15
easyIs ?
Example 16
challengeShow that is the only set with no proper supersets among its own subsets — i.e. iff ? Evaluate this claim.
Example 17
easyWhat is for any set ?
Example 18
easyDetermine whether each set is empty: (a) , (b) , (c) .
Example 19
mediumLet . Find: (a) for any set , (b) for any set , (c) (the power set of ).
Example 20
challengeProve that the empty set is unique (there is only one empty set).