Calculating Proper Subsets for Set Operations

The problem asks us to find the number of proper subsets of a set S, which is defined based on operations involving three distinct sets X, Y, and Z with given cardinalities.

We are given the following information:

• Set X has 6 elements, so $|X| = 6$.
• Set Y has 5 elements, so $|Y| = 5$.
• Set Z has 4 elements, so $|Z| = 4$.
• All 15 elements across sets X, Y, and Z are distinct. This is a critical piece of information.

The set S is defined as $S = (X - Y) \cup Z$. We need to find the number of proper subsets of S.

Determining the Cardinality of Set S

First, let's find the number of elements in set S, which is denoted by $|S|$. The definition of S involves two set operations: set difference $(X - Y)$ and set union $(\cup)$ with Z.

Step 1: Evaluate the Set Difference (X - Y)

The set $(X - Y)$ contains elements that are in X but not in Y. Since all elements in X, Y, and Z are distinct, there is no overlap between set X and set Y. This means $X \cap Y = \emptyset$.

Therefore, the set of elements in X that are not in Y is simply the set X itself, because no element of X is in Y.

So, $(X - Y) = X$.

The cardinality of $(X - Y)$ is $|X - Y| = |X| = 6$.

Step 2: Evaluate the Set Union (X - Y) $\cup$ Z

Now we need to find the cardinality of $S = (X - Y) \cup Z$. We have established that $(X - Y) = X$. So, $S = X \cup Z$.

To find the cardinality of the union of two sets, we use the formula: $|A \cup B| = |A| + |B| - |A \cap B|$.

In our case, $A = X$ and $B = Z$. So, $|S| = |X \cup Z| = |X| + |Z| - |X \cap Z|$.

We know $|X| = 6$ and $|Z| = 4$. We need to find $|X \cap Z|$.

The problem states that all 15 elements across sets X, Y, and Z are distinct. This implies that there is no overlap between X and Z either. The intersection of X and Z is an empty set, $X \cap Z = \emptyset$.

Therefore, $|X \cap Z| = 0$.

Now, substitute the values into the union formula:

$|S| = |X| + |Z| - |X \cap Z|$

$|S| = 6 + 4 - 0$

$|S| = 10$

So, the set S has 10 elements.

Calculating the Number of Proper Subsets

A set with 'n' elements has a total of $2^n$ subsets. A proper subset is any subset of a set except the set itself.

The number of proper subsets is given by the formula: (Total number of subsets) - 1.

Number of proper subsets $= 2^{|S|} - 1$.

Since $|S| = 10$, the number of proper subsets is $2^{10} - 1$.

We know that $2^{10} = 1024$.

Number of proper subsets $= 1024 - 1 = 1023$.

Therefore, set S has 1023 proper subsets.

Total number of subsets of S $= 2^{|S|} = 2^{10} = 1024$.

Number of proper subsets of S $= 2^{|S|} - 1 = 1024 - 1 = 1023$.

Revision Table: Key Concepts

Additional Information: Understanding Distinct Elements

The condition that all elements across X, Y, and Z are distinct simplifies the calculation significantly. If the elements were not distinct, we would need more information about the intersections ($X \cap Y$, $X \cap Z$, $Y \cap Z$, and $X \cap Y \cap Z$) to accurately calculate $|X - Y|$ and $|(X - Y) \cup Z|$.

• If elements were not distinct, $(X - Y)$ would contain elements of X that are not also present in Y. The size would be $|X| - |X \cap Y|$.
• Similarly, the union of $(X - Y)$ and Z would require knowing the intersection of $(X - Y)$ and Z: $|(X - Y) \cup Z| = |X - Y| + |Z| - |(X - Y) \cap Z|$.

However, in this specific problem, the distinctness of all 15 elements guarantees that any pair of sets from X, Y, Z have an empty intersection. This simplifies $(X-Y)$ to X (since $X \cap Y = \emptyset$) and $(X-Y) \cup Z$ to $X \cup Z$ (since $X \cap Z = \emptyset$).