Ryan K. answered • 09/22/20
Math, Music, and Multi-Subject Tutor from Georgia Tech
Hi Hasaan!
I'll start with some definitions since I don't know your background.
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∪ : This symbol is called "union" or "or." If A is a set and B is a set, A ∪ B is called the union of A and B. A ∪ B is the set of everything that is in A or in B or both.
Facts:
A ∪ B = B ∪ A .
(A ∪ B) ∪ C = A ∪ (B ∪ C)
∩ : This symbol is called "intersection" or "and." If A is a set and B is a set, A ∩ B is called the intersection of A and B. A ∩ B is the set of everything that is in both A and B.
Facts:
A ∩ B = B ∩ A .
(A ∩ B) ∩ C = A ∩ (B ∩ C)
A' : The ' symbol is called "negation" or "not." If A is a set, A' is called the negation of A. A' is the set of everything that is not in A.
Fact: (A')' = A .
U : A little confusing because it looks similar to ∪. U is the universal set, or all possible elements.
Facts:
A ∪ U = U
A ∩ U = A
A ∪ A' = U
∅ : The empty set, which contains no elements.
Facts:
A ∪ ∅ = A
A ∩ ∅ = ∅
A ∩ A' = ∅
U' = ∅
De Morgan's Laws: There are two of these: 1. (A ∪ B)' = A' ∩ B' and 2. (A ∩ B)' = A' ∪ B'
(A ∪ B)' = A' ∩ B' : The left side, (A ∪ B)', is everything that is not in the set A ∪ B.
Since everything in A and everything in B are in A ∪ B, the only things not in A ∪ B are the things that are not in A and not in B.
That's exactly what the right side is: A' ∩ B' is everything that is not in the set A and not in the set B.
(A ∩ B)' = A' ∪ B' : The left side, (A ∩ B)', is everything that is not in the set A ∩ B.
Since the only things in A ∩ B are the things that are in both A and B, everything that is not in A or not in B will not be in A ∩ B.
That's exactly what the right side is: A' ∪ B' is everything that is either not in the set A or not in the set B or in neither set A nor set B.
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Now for your questions.
(a) (S′ ∩ T)′ ∪ T
Take (S′ ∩ T)′ first. De Morgan's Laws tell us that (A ∩ B)' = A' ∪ B' . We can replace A with S' and replace B with T to get:
(S′ ∩ T)′ = (S')' ∪ T' = S U T' .
(S′ ∩ T)′ ∪ T = (S')' ∪ T' U T= S ∪ T' ∪ T = S ∪ (T' ∪ T) = S ∪ U = U
The answer is U, the universal set.
(b) (S′ ∪ T′)′
De Morgan's Laws tell us that (A ∪ B)' = A' ∩ B' . We can replace A with S' and replace B with T' to get:
(S′ ∪ T′)′ = (S')' ∩ (T')' = S ∩ T .
The answer is S ∩ T .
(c) T ∩ (S ∪ T)′
Take (S ∪ T)′ first. De Morgan's Laws tell us that (A ∪ B)' = A' ∩ B' . We can replace A with S and replace B with T to get:
(S ∪ T)′ = S' ∩ T'
T ∩ (S ∪ T)′ = T ∩ S' ∩ T' = T ∩ T' ∩ S = (T ∩ T') ∩ S = ∅ ∩ S = ∅
The answer is ∅, the empty set.
You may find this written answer helpful, but there are many other ways to explain. Feel free to reach out to me to schedule a lesson if you have more questions.
