Question

Let a, b, and c be sets. show that

a) (a ∪ b) ⊆ (a ∪ b ∪ c).
b) (a ∩ b ∩ c) ⊆ (a ∩ b).
c) (a − b) − c ⊆ a − c.
d) (a − c) ∩ (c − b) = ∅.
e) (b − a) ∪ (c − a) = (b ∪ c) − a

Answer 1

The student's question pertains to set theory and involves demonstrating the relationship between various set operations such as unions, intersections, and differences. Each statement highlights different properties of sets in relation to subset inclusion and equality.

Step-by-step explanation:

To illustrate the set theory concepts presented in the question, let's examine each statement separately:

1. (a ∪ b) ⊆ (a ∪ b ∪ c): This statement means every element in the union of sets a and b is also in the union of a, b, and c. This is because adding additional elements to a set does not remove any existing elements.
2. (a ∩ b ∩ c) ⊆ (a ∩ b): This means the intersection of sets a, b, and c is a subset of the intersection of a and b. This is true because the intersection of a, b, and c contains only elements common to all three sets, which are also common to just a and b.
3. (a − b) − c ⊆ a − c: This explains that if we remove elements of b from a, and then remove elements of c from what remains, the result is a subset of simply removing c from a. This is because once elements in b are removed, subtracting elements in c cannot reintroduce elements from b.
4. (a − c) ∩ (c − b) = ∅: It is stated that the intersection of the set difference a-c and c-b is empty. This is true as (a-c) means all elements in a but not in c, and (c-b) means all elements in c but not in b; therefore, they cannot have any elements in common.
5. (b − a) ∪ (c − a) = (b ∪ c) − a: This final relationship indicates that the union of set differences b-a and c-a (elements in b or c but not in a) is equivalent to the set difference of the union of b and c with a.