Description

1. 5 points Write down P(X) if X = {∅,{α,β,γ},γ,{{α,β}}}.
Solution: P(X) = {∅,{∅},{{α,β,γ}},{γ},{{{α,β}}},{∅,{α,β,γ}},{∅,γ},{∅,{{α,β}}},
{{α,β,γ},γ},{{α,β,γ},{{α,β}}},{γ,{{α,β}}},{∅,{α,β,γ},γ},{∅,{α,β,γ},{{α,β}}},
{∅,γ,{{α,β}}},{{α,β,γ},γ,{{α,β}}},{∅,{α,β,γ},γ,{{α,β}}}}
Solution: F = {x | x is a DSSE faculty member }
C = {x | x is a DSSE course }
Possible pairs can be made through Cartesian Product between F and C S = F x C
S = {x ∈ (f,c) | f ∈ F ∧ c ∈ C}
(b) 5 points Imagine that the the operation above is extended to include an additional set that contains all the time slots when a course can be scheduled. Describe the set obtained as an outcome of the operation.
Solution: T = {x | x is a time slot when a course can be scheduled }
To include time slots as well, Cartesian Product will be taken between S and T P = S x T
P = {x ∈ (f,c,t) | f ∈ F ∧ c ∈ C ∧ t ∈ T}
The set P contains all possible combinations between DSSE faculty members, and DSSE courses, and the time slots in which a course can be scheduled.
3. The symmetric difference of two sets A and B is defined as
A ⊕ B = (A − B) ∪ (B − A).
1
It is also known as the disjunctive union as it contains all those elements which are in either of those sets, but not in their intersection.

A ⊕ B ⊕ C = (A ⊕ B) ⊕ C,
i.e. the two-set definition is applied twice. Draw the Venn diagram of this set.

(c) 5 points Using the insights from above, express A⊕B ⊕C in the same manner as given in part a). That is, using the basic set operations: union, intersection, and complement. Show your working.

4. Let A be the set of all numbers that are divisible by 6 and B the set of all numbers that are divisible by 10.
(a) 5 points Write the sets A and B in set notation and describe A∩B as simply as possible.
Solution: A = {6x | x ∈ N}
B = {10x | x ∈ N}
A ∩ B is a set with all numbers divisible by 30.
A ∩ B = {30x | x ∈ N}
(b) 10 points Describe the set A ⊕ B, i.e. the symmetric difference of A and B, using set notation. Provide a proof that the set you indicate is indeed the symmetric difference of A and B.
Solution: A ⊕ B is a set with numbers divisible by 6 or 10 but not both.
=⇒ A ⊕ B = {x | (x ∈ A ∧ x /∈ B) ∨ (x ∈ B ∧ x /∈ A)}
Let x ∈ A ⊕ B
Then, (x ∈ A and x /∈ B) or (x ∈ B and x /∈ A)

That is, x ∈ ((A ∩ B) ∪ (B ∩ A))
Then by definition of set complement, x ∈ ((A − B) ∪ (B − A))
Since A ⊕ B = (A − B) ∪ (B − A), and x ∈ (A − B) ∪ (B − A), therefore proved that x ∈ A ⊕ B and the indicated set is the symmetric difference of A and B.
(c) 5 points Given U = {x ∈ N | x ≤ 60}, list the elements of A, B, and A ⊕ B
Solution: A = {6,12,18,20,24,30,36,42,48,54,60}
B = {10,20,30,40,50,60}
A ⊕ B = {6,10,12,18,20,24,36,40,42,48,50,54}

5. Show that A ∪ B = A ∩ B.
(a) 5 points by using set identities.
Solution: Starting with LHS:

A ∪ B = A ∩ (B) DeMorgan’s Law

= A ∩ B
Hence proved using set identities Law of Double Complementation
(b) 5 points by proving that each set is a subset of the other.