Question

If \(\mathrm{X}, \mathrm{Y}\), and \(\mathrm{Z}\) are any three subsets of the universal set \(\mathrm{U}\), prove that $$ (\mathrm{X} \cap \mathrm{Y})-(\mathrm{X} \cap \mathrm{Z})=\mathrm{X} \cap(\mathrm{Y}-\mathrm{Z}) $$

Short answer

To prove \((X \cap Y) - (X \cap Z) = X \cap (Y - Z)\), we define two sets: \(A = (X \cap Y) - (X \cap Z)\) and \(B = X \cap (Y - Z)\). We then show that \(A\) is a subset of \(B\) and \(B\) is a subset of \(A\). Since both sets are subsets of each other, it can be concluded that \(A = B\), proving the given statement.

Step-by-step solution

Understanding Set Notations

The notation \(X \cap Y\) denotes the intersection of sets X and Y - this is a set including the elements that are common to both X and Y. On the other hand, \(Y - Z\) represents the difference of set Z from set Y, which results in a set that includes all the elements in Y that are not in Z.

Defining the sets

Define the set \(A = (X \cap Y) - (X \cap Z)\) and the set \(B = X \cap (Y - Z)\). The goal now is to prove \(A = B\), which can be done by showing that \(A\) is a subset of \(B\) and \(B\) is a subset of \(A\).

Proving A is a subset of B

Pick an arbitrary element \(x\) from set \(A\). This means that \(x\) lies in the intersection of X and Y and not in the intersection of X and Z. Hence \(x\) belongs to both X and Y, but not Z. Therefore \(x\) is in X and in the set \(Y - Z\) as well, which means \(x\) belongs to \(B\). This shows that every element in \(A\) is in \(B\) i.e., \(A\) is a subset of \(B\).

Proving B is a subset of A

Similarly, pick an arbitrary element \(y\) from set \(B\). This means \(y\) belongs to X and also to the set that results from removing Z from Y. Hence \(y\) belongs to both X and Y and not to Z. Therefore \(y\) is in the set \(Y - Z\) and in X as well, which means \(y\) belongs to \(A\). This shows that every element in \(B\) is in \(A\) i.e., \(B\) is a subset of \(A\).

Conclusion

Since \(A\) is a subset of \(B\) and \(B\) is a subset of \(A\), it can be finalized that \(A\) is equal to \(B\). Hence, \((X \cap Y) - (X \cap Z) = X \cap (Y - Z)\) is proven. This completes the exercise.

Key concepts

Intersection of Sets

In set theory, the intersection of sets represents the common elements that two or more sets share. Imagine having two collections of objects; the intersection would be like finding the items they both have. Mathematically, when we talk about the intersection of sets X and Y, denoted as \(X \cap Y\), we're looking specifically at all the elements that set X and set Y both contain.

For example, if set \(X = \{1, 2, 3\}\) and set \(Y = \{2, 3, 4\}\), their intersection \(X \cap Y\) would be \(\{2, 3\}\) because those are the numbers that appear in both sets. This concept is pivotal in our exercise because, in the given proof, we're employing the intersection to filter elements that are common to sets \(X\) and \(Y\) but then refining that set further by excluding certain elements.

Set Difference

The set difference is another fundamental concept in set theory. It represents the elements that are in one set but not in another. For instance, when we describe set difference \(Y - Z\), it means we take all the elements in set Y and remove any that are also in set Z. What's left is the set difference.

Continuing with our previous sets, if \(Z = \{3, 4, 5\}\), the set difference \(Y - Z\) would be \(\{2\}\) because that's the element that is in Y but not in Z. This operation is crucial in the solution we're discussing as it helps to isolate elements of \(Y\) that are not shared with \(Z\), which is then intersected with set \(X\) for the final proof.

Subset Proof

Subsets are a way to describe the relationship between two sets. Formally, set A is a subset of set B if every element of A is also an element of B. In proving that one set is a subset of another, we typically take an arbitrary element from the first set and show that it must also be an element of the second set.

In our exercise, the subset proof is employed twice to show that set \(A\), which is \((X \cap Y) - (X \cap Z)\), and set \(B\), which is \(X \cap (Y - Z)\), are equal by demonstrating that each is a subset of the other. This two-way subset relationship establishes the equality of the two sets, which is the heart of this exercise's proof.

Universal Set

In the realm of set theory, the universal set \(U\) is a set that includes all the possible elements under consideration and is the 'universe' with respect to which other sets are defined. Every other set in this context is a subset of the universal set.

In our particular problem, the sets \(X\), \(Y\), and \(Z\) are presumed to be subsets of this universal set \(U\). Acknowledging the existence of a universal set is essential because it provides a reference for all elements that could potentially be in our sets of interest. It is the backdrop against which we perform our operations like intersection and set difference, and without it, the scope of our elements would be undefined.