Math 243

Each set operator corresponds to a logical operator, in the following sense: for each binary set operator □, there is a binary logical operator ○ such that for any universal set U, and any subsets A and B of U, A □ B = { x ∈ U | x ∈ A ○ x ∈ B }.Match set operators and their corresponding logical operators. 1.difference 2.union 3.intersection 4.symmetric difference

1. and not 2. or 3. and 4. exclusive or

Check all true statements. 1.For all sets A, B, the power set of the cartesian product of A and B is the cartesian product of the power sets of A and B. 2.The cartesian product has a cancellation property, i.e. if A x B = A x C for some sets A, B, C and A nonempty, then B = C. 3.For all nonempty sets A, B, C, D, if A x B = C x D, then A = C and B = D. 4.The cartesian product has a cancellation property, i.e. if A x B = A x C for some sets A, B, C, then B = C. 5.The cartesian product is associative, i.e. (A x B) x C = A x (B x C) =for all sets A, B, C. Both sets are equal to A x B x C. 6.For all sets A, B, the cardinality of A x B is the sum of the cardinalities of A and B. 7.If A={1,2} and B={3} then A x B ={ (1,3), (2,3) }. 8.If A={1,2} and B={3} then A x B ={ {1,3}, {2,3} }. 9.The cartesian product is symmetric, i.e. A x B = B x A for all sets A, B.

1.For all nonempty sets A, B, C, D, if A x B = C x D, then A = C and B = D. 2.The cartesian product has a cancellation property, i.e. if A x B = A x C for some sets A, B, C and A nonempty, then B = C. 3.If A={1,2} and B={3} then A x B ={ (1,3), (2,3) }.

Mark all statements that are correct. Group of answer choices 1.The empty set is a proper subset of every set. 2.The empty set, and only the empty set, has cardinality zero. 3.The empty set has no subsets. 4.For any universal set, the complement of the empty set is nonempty. 5.A union with the empty set is always empty. 6.The power set of the empty set is empty. 7.The empty set is a subset of every set. 8.The cartesian product A × B is empty if A is empty or B is empty. 9.An intersection with the empty set is always empty. 10.The empty set has no proper subsets.

1.The empty set, and only the empty set, has cardinality zero. 2.The empty set is a subset of every set. 3.The cartesian product A × B is empty if A is empty or B is empty. 4.An intersection with the empty set is always empty. 5.The empty set has no proper subsets.

Mark the true statements. Group of answer choices 1.The set { {} } is not empty. 2.The power set of the power set of the empty set is { ∅ , ∅ } 3{{1,2}} is an element of the power set of {1,2,3}. 4.{1,2} is an element of the power set of {1,2,3}. 5.The power set of the power set of the empty set is { { ∅, { ∅ } } }. 6.The power set of the power set of the empty set is { ∅, { ∅ } }. 7.{{1,2}} is a subset of the power set of {1,2,3}. 8.The power set of the empty set is { ∅ }. 9.{1,2} is a subset of the power set of {1,2,3}. 10.The power set of the empty set is the empty set.

1.The set { {} } is not empty. 2.{1,2} is an element of the power set of {1,2,3}. 3.The power set of the power set of the empty set is { ∅, { ∅ } }. 4.{{1,2}} is a subset of the power set of {1,2,3}. 5.The power set of the empty set is { ∅ }.

All the binary set operators we learned in the class are symmetric, except one. Identify the set operator that is not symmetric. (a binary set operator □ is called symmetric if A □ B = B □ A for all universal sets U, and all subsets A,B of U.)

1.difference

True or false? ∅ and {∅} are the same set. They are both the empty set.

False