Question

Show that $$ A-(B \cup C)=(A-B) \cap(A-C), \quad A-(B \cap C)=(A-B) \cup(A-C) $$

Short answer

The provided identities are true, as by utilizing the laws and definitions of set operations (difference, union, intersection) you can match the descriptions of elements belonging to both sides of the equal signs in provided identities.

Step-by-step solution

Expressing first identity

To prove \(A-(B \cup C)=(A-B) \cap(A-C)\), remember that an element belongs to a difference of two sets if it is in the first set and not in the second. Therefore, you can tell that an element belongs to \(A-(B \cup C)\) if and only if it is in set A and not in \(B \cup C\). According to the definition of the union operation, it means that the element is not in B neither in C. This description matches with the description of an element belonging to the set \((A-B) \cap(A-C)\). Hence, \(A-(B \cup C)=(A-B) \cap(A-C)\)

Expressing second identity

To prove \(A-(B \cap C)=(A-B) \cup(A-C)\), apply the same logic about the difference of sets. An element will belong to \(A-(B \cap C)\) if it is in A and not simultaneously in B and C (because B and C make an intersection). This matches the description of the union operation stating that element should be in A but not in B, or in A but not in C. This description fits with an element belonging to \((A-B) \cup(A-C)\). Hence, \(A-(B \cap C)=(A-B) \cup(A-C)\)

Key concepts

Set Difference

In set theory, the concept of set difference is fundamental. Imagine you have two sets, say, set \( A \) and set \( B \). The set difference, denoted by \( A - B \), refers to the elements that are in set \( A \) but not in set \( B \). This is akin to removing all elements of \( B \) from \( A \).
For instance, if \( A = \{1, 2, 3, 4\} \) and \( B = \{3, 4, 5\} \), the set difference \( A - B \) is \( \{1, 2\} \) because 1 and 2 are in \( A \) but not in \( B \).
Understanding set difference helps in distinguishing elements that are unique to a particular set when compared with another. This concept plays a crucial role in solving various set-theoretical problems.

Union of Sets

The union of sets captures the idea of combining elements. When you take the union of two sets, you are essentially gathering all the distinct elements from both sets into a new set.
This operation is denoted as \( A \cup B \). For example, if \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), the union \( A \cup B \) results in \( \{1, 2, 3, 4, 5\} \). Notice how any repeated elements are only listed once.
Union is a straightforward but powerful operation. It helps in pooling resources by combining elements from separate sets and is useful in scenarios where one wishes to draw from all options available.

Intersection of Sets

Intersection of sets represents the common ground between two sets. With intersection, you identify the elements that appear in both sets.
The intersection is represented as \( A \cap B \). For instance, given \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), the intersection \( A \cap B \) is \( \{2, 3\} \) because these are the only elements present in both sets.
Understanding this concept is important because many problems in mathematics and real-world applications require finding shared characteristics or common factors between different groups.

Proof in Set Theory

Proving identities and relationships is an integral part of set theory. Utilizing logical reasoning with known operations like union, intersection, and difference allows one to establish mathematical truths.
For example, with the identity \( A-(B \cup C)=(A-B) \cap(A-C) \), a proof involves showing that any element in \( A-(B \cup C) \) must logically be in \( A-B \) and \( A-C \) separately, thus intersecting at those elements neither in \( B \) nor in \( C \).
Similarly, for \( A-(B \cap C)=(A-B) \cup(A-C) \), one must prove that any element part of \( A \) and not of \( B \cap C \) can only be avoided in \( B \), \( C \), or both, leading to elements in \( (A-B) \cup(A-C) \).
Proofs not only validate results but also deepen understanding by revealing the interconnected nature of mathematical concepts.