3. A company offers two services: mobile banking and SMS alerts. Out of 100 clients, 65 subscribe to mobile banking, 50 to SMS alerts, and 30 to both services.
(a) How many clients subscribe to at least one of the two services?
(b) How many clients subscribe to neither service?
4. Let [tex]$S=\{a, b, c\}$[/tex].
(a) List all elements of the power set [tex]$P (S)$[/tex].
(b) What is the cardinality of [tex]$P (S)$[/tex]?
Answer (1)
Use inclusion-exclusion principle: $|A
B| = |A| + |B| - |A
B| = 65 + 50 - 30 = 85$.
Clients subscribing to neither service: 100 − 85 = 15 .
List all subsets of S = { a , b , c } : ∅ , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c } .
Cardinality of P ( S ) : 2 3 = 8 .
Explanation
Problem Overview and Strategy Let's break down this problem step by step. We'll start by finding out how many clients subscribe to at least one of the services offered by the company. Then, we'll determine how many clients don't subscribe to any of the services. Finally, we'll explore the concept of a power set using the given set S.
Clients Subscribing to At Least One Service To find the number of clients subscribing to at least one service, we'll use the principle of inclusion-exclusion. This principle helps us avoid double-counting clients who subscribe to both services. The formula is:
$|A
B| = |A| + |B| - |A
B|$
Where:
∣ A ∣ is the number of clients subscribing to mobile banking (65).
∣ B ∣ is the number of clients subscribing to SMS alerts (50).
$|A
B|$ is the number of clients subscribing to both services (30).
Let's plug in the values:
$|A
B| = 65 + 50 - 30 = 85$
So, 85 clients subscribe to at least one service.
Clients Subscribing to Neither Service Now, to find the number of clients subscribing to neither service, we subtract the number of clients subscribing to at least one service from the total number of clients (100):
100 − 85 = 15
Therefore, 15 clients subscribe to neither service.
Listing Elements of the Power Set P(S) Next, let's explore the power set P ( S ) of the set S = { a , b , c } . The power set is the set of all possible subsets of S , including the empty set and S itself. Here are all the elements of P ( S ) :
∅ (the empty set)
{ a }
{ b }
{ c }
{ a , b }
{ a , c }
{ b , c }
{ a , b , c }
Cardinality of the Power Set P(S) Finally, the cardinality of the power set P ( S ) is the number of elements in P ( S ) . Since S has 3 elements, the cardinality of P ( S ) is 2 3 = 8 . We can also count the elements we listed in the previous step to confirm this.
Final Answers In summary:
(a) 85 clients subscribe to at least one of the two services.
(b) 15 clients subscribe to neither service.
(c) The power set P ( S ) = { ∅ , { a } , { b } , { c } , { a , b } , { a , c } , { b , c } , { a , b , c }} .
(d) The cardinality of P ( S ) is 8.
Examples
Understanding sets and probabilities is very useful in real life. For example, if you're analyzing customer preferences for different product features, you can use set theory to understand how many customers like certain combinations of features. Similarly, power sets are used in computer science to represent all possible combinations of settings or options in a program.
