CmSc 365 Theory of Computation
Solutions to Exercise on Sets
1.
Subsets A ⊆ B
True or false?
{1,2,3} ⊆ {2,3,4}
False
{1} ⊆ {1,2,3}
True
{1} ⊆ {{1},1, 2, 3}
True
{1} ⊆ {{1}, 2, 3}
False
Ø ⊆ {1, 2}
True
Ø ⊆ Ø
True
Ø ⊆ {Ø}
True
{1,2,3} є {2,3,4}
False
{1} є {1,2,3}
False
{1} є {{1},1, 2, 3}
True
{1} є {{1}, 2, 3}
True
{1} є {{1,2}, 2, 3}
False
1
є {{1,2}, 2, 3}
False
2
є {{1,2}, 2, 3}
True
{1,2} є {{1,2}, 2, 3}
True
Ø є {1, 2}
False
Ø є Ø
False
Ø є {Ø}
True
2. Operations on sets
a. Intersections A ∩ B = {x : (x є A ) Λ (x є B)}
Define the intersection of the set containing all positive numbers and the set of all
numbers divisible by 11 using the setbuilder notation
A = { x : x > 0 Λ ∃ q such that q > 0 Λ x = 11*q}
b. Unions A ∪ B = {x : (x є A) V (x є B)}
A⊆A∪B, B ⊆ A∪B
A ∪ Ø = A, A ∪ U = U
Define the union of the set containing all positive numbers and the set of all numbers
divisible by 11 using the setbuilder notation
A = { x : x > 0 V ∃ q such that q > 0 Λ x = 11*q}
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c. Differences A - B = { x : ( x є A) Λ ( x ∉ B)}
A-Ø=A
A -U=Ø
Define the difference of the set containing all positive numbers and the set of all numbers
divisible by 11 using the setbuilder notation
A = { x : x > 0 Λ ∼(∃ q such that q > 0 Λ x = 11*q) } =
{ x : x > 0 Λ ∀ q > 0, x ≠ 11*q) }
Will 22 be in the difference?
Will -22 be in the difference?
Will 21 be in the difference?
No
No
Yes
Complementary set of a set A: ~A = U - A
Define the compliment of the set containing all positive numbers divisible by 11 using
the setbuilder notation
A = { x : ~(x > 0 Λ ∃ q such that q > 0 Λ x = 11*q)} =
{ x : x ≤ 0 V ∀ q > 0, x ≠ 11*q}
Will 22 be in the complement? No
Will -22 be in the complement? Yes
Will 21 be in the complement? Yes
3. Basic set identities
Prove that A - (B ∩ C)
A - (B ∩ C) =
=
=
=
= (A - B) ∪ (A - C)
A ∩ ~ (B ∩ C)
Alternate representation of set difference
A ∩ (~B ∪ ~C)
By De Morgan's Laws
(A ∩ ~B ) ∪ (A ∩ ~C) By distributive laws
(A - B) ∪ (A – C)
Alternate representation of set difference
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4. Disjoint sets, Cartesian product, Power sets, Partitions
Disjoint sets: S1 and S2 are disjoint sets iff S1 ∩ S2 = ∅
Are the sets A = { 1,2,{3}} and B = {1,2,3} disjoint?
No
Cartesian product of two sets A and B: A x B = {(x,y) : x є A Λ y є B}
A x B x C = {(x,y,z): x є A Λ y є B Λ z є C}
Find the Cartesian product of {∅} and {a,b}
{ (∅, a), (∅,b)}
Find the Cartesian product of ∅ and {a,b}
∅
Power set of a set A: 2A The set of all subsets of A
Find 2{1, 2, 3}
2{1, 2, 3} = { ∅ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
Find 2{1, 2, {3}}
2{1, 2, {3}} = { ∅ , {1}, {2}, {{3}}, {1,2}, {1,{3}}, {2,{3}}, {1,2,{3}}}
Find 2{1, ∅ }
2{1, ∅ } = { ∅ , {1}, {∅}, {1,∅}}
Partition of a nonempty set A: A non-empty set Π ⊂ 2 A, such that
(1) Ø ∉ Π
(2) No two distinct sets in Π have common elements
(3) U Π = A
Find two partitions of A = {1, 2, {1}}
{{1}, {2, {1}}}, {{1,{1}}, {2}}
Find the Cartesian product C of A = {1,2} and B = {a,b,c}
C = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}
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