2. a) Answer the following questions.
[tex]$[1+1]$[/tex]
i) Is [tex]$A =\{1,3,5,7,9\}$[/tex] a subset of [tex]$B =\{1,2,3,4,5, \ldots \ldots, 10\}$[/tex]? Give reason.
ii) Is [tex]$P =\{1,2,4,5,10,20\}$[/tex] a proper or improper subset of [tex]$Q=\{x: x$[/tex] is a factor of 20[tex]$\}\[/tex]? Give reason.
b) Let [tex]$A=\{s, e, t\}$[/tex] is a set of letters of the word 'set'
[tex]$[1+1+2]$[/tex]
i) Write the cardinal numbers of the set A.
ii) Find how many subsets of the set A are possible?
iii) Write all possible subsets of the set [tex]$A$[/tex].
Answer (1)
A is a subset of B because all elements in A are also in B .
P is an improper subset of Q because P and Q are equal.
The cardinal number of set A = { s , e , t } is 3, meaning it has 3 elements.
Set A has 2 3 = 8 possible subsets: { } , { s } , { e } , { t } , { s , e } , { s , t } , { e , t } , { s , e , t } .
The final answers are: a) i) Yes ii) Improper subset b) i) 3 ii) 8 iii) { } , { s } , { e } , { t } , { s , e } , { s , t } , { e , t } , { s , e , t } .
A ⊆ B , P is improper subset of Q , ∣ A ∣ = 3 , Number of subsets = 8 , Subsets of A are { } , { s } , { e } , { t } , { s , e } , { s , t } , { e , t } , { s , e , t }
Explanation
Checking if A is a subset of B We are given two sets, A = { 1 , 3 , 5 , 7 , 9 } and B = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } . We need to determine if A is a subset of B . A set A is a subset of B if every element in A is also an element in B .
Verifying elements of A in B We examine each element of A to see if it is also in B . The elements of A are 1, 3, 5, 7, and 9. All of these elements are also in B . Therefore, A is a subset of B .
Determining if P is a proper or improper subset of Q We are given two sets, P = { 1 , 2 , 4 , 5 , 10 , 20 } and Q = { x : x is a factor of 20 } . We need to determine if P is a proper or improper subset of Q . First, let's list the elements of Q . The factors of 20 are 1, 2, 4, 5, 10, and 20. So, Q = { 1 , 2 , 4 , 5 , 10 , 20 } .
Comparing P and Q Now we compare the sets P and Q . We see that P = { 1 , 2 , 4 , 5 , 10 , 20 } and Q = { 1 , 2 , 4 , 5 , 10 , 20 } . Since P = Q , P is an improper subset of Q . An improper subset is a subset that is equal to the original set.
Finding the cardinal number of A We are given the set A = { s , e , t } . We need to find the cardinal number of A . The cardinal number of a set is the number of elements in the set. In this case, A has 3 elements: s , e , and t . Therefore, the cardinal number of A is 3.
Finding the number of subsets of A We need to find the number of subsets of the set A = { s , e , t } . The number of subsets of a set with n elements is 2 n . Since A has 3 elements, the number of subsets of A is 2 3 = 8 .
Listing all possible subsets of A We need to list all possible subsets of the set A = { s , e , t } . The subsets are:
The empty set: { }
Subsets with one element: { s } , { e } , { t }
Subsets with two elements: { s , e } , { s , t } , { e , t }
Subset with three elements: { s , e , t }
Final Answer Therefore, the answers are: a) i) Yes, A is a subset of B because all elements of A are in B .
ii) P is an improper subset of Q because P = Q .
b) i) The cardinal number of A is 3. ii) The number of subsets of A is 8. iii) The subsets of A are { } , { s } , { e } , { t } , { s , e } , { s , t } , { e , t } , { s , e , t } .
Examples
Understanding subsets is crucial in database management. For example, if you have a database of all customers (set B), a subset (set A) might be the customers who purchased a specific product. Identifying subsets helps in targeted marketing and data analysis. Similarly, in genetics, the set of all genes in an organism can be analyzed to identify subsets responsible for specific traits.
