Understanding the Faculty Connectivity Problem

This problem involves using set theory concepts to determine the number of faculty members with only a specific type of social media account. We are given the total number of faculty members, the number connected via Facebook, the number via WhatsApp, and the number with neither account. We need to find the number connected only through Facebook.

Defining the Sets and Given Data

Let's define the sets involved:

• Total number of faculty members, $|U| = 150$
• Number of faculty members with Facebook accounts, $|F| = 55$
• Number of faculty members with WhatsApp accounts, $|W| = 85$
• Number of faculty members with neither account, $|(F \cup W)'| = 30$

We want to find the number of faculty members connected only through Facebook accounts, which can be represented as $|F \text{ only}|$.

Calculating Faculty Members with At Least One Account

The total number of faculty members is the sum of those with at least one account (Facebook or WhatsApp or both) and those with neither account. Using the principle of inclusion-exclusion for the total set:

\(|U| = |F \cup W| + |(F \cup W)'|\)

We can find the number of faculty members with at least one account:

\(|F \cup W| = |U| - |(F \cup W)'|\)

Substituting the given values:

\(|F \cup W| = 150 - 30\)

\(|F \cup W| = 120\)

So, 120 faculty members are connected through Facebook or WhatsApp or both.

Finding Faculty Members with Both Accounts

The formula for the union of two sets is:

\(|F \cup W| = |F| + |W| - |F \cap W|\)

Here, $|F \cap W|$ represents the number of faculty members connected through both Facebook and WhatsApp accounts.

We know $|F \cup W| = 120$, $|F| = 55$, and $|W| = 85$. Substituting these values into the formula:

\(120 = 55 + 85 - |F \cap W|\)

\(120 = 140 - |F \cap W|\)

Now, we can solve for $|F \cap W|$:

\(|F \cap W| = 140 - 120\)

\(|F \cap W| = 20\)

Therefore, 20 faculty members are connected through both Facebook and WhatsApp.

Calculating Faculty Members Connected Only Through Facebook

The number of faculty members connected only through Facebook is the total number of Facebook users minus those who are also connected through WhatsApp (since those are included in the total Facebook count).

\(|F \text{ only}| = |F| - |F \cap W|\)

We know $|F| = 55$ and $|F \cap W| = 20$. Substituting these values:

\(|F \text{ only}| = 55 - 20\)

\(|F \text{ only}| = 35\)

Thus, 35 faculty members are connected only through Facebook accounts.

Summary of Results

The number of faculty members connected only through Facebook accounts is 35.

Revision Table: Key Set Theory Concepts

Additional Information: Visualizing with Venn Diagrams

While not required for the calculation, visualizing this problem with a Venn diagram can be helpful. Draw two overlapping circles, one for Facebook (F) and one for WhatsApp (W), within a rectangle representing the Universal Set (U). The overlapping area is the intersection (\(F \cap W\)). The area inside the F circle but outside the overlap is Only Facebook (\(F \text{ only}\)). The area inside the W circle but outside the overlap is Only WhatsApp (\(W \text{ only}\)). The area outside both circles but inside the rectangle is Neither (\((F \cup W)'\)).

• The sum of the areas for Only F, Only W, and Both F&W equals \(|F \cup W|\).
• The sum of Only F and Both F&W equals \(|F|\).
• The sum of Only W and Both F&W equals \(|W|\).
• The sum of Only F, Only W, Both F&W, and Neither equals \(|U|\).

Using our calculated values:

• Both (F & W) = 20
• Only F = 35 (calculated)
• Only W = 65 (calculated as \(85 - 20\))
• Neither = 30 (given)

Check the total: \(35 + 65 + 20 + 30 = 150\), which matches the total number of faculty members.