Today is our topic of discussion Set example.
Set example
Example 1.
If A = {x: x is a positive integer }, B = {0} and X = {x: x is an integer), then what is the relationship between A, B and X?
Solution: Here, A⊆X, B⊆X, B⊄A.
Example 2.
{x: x is a real number and x² < 0} is an empty set, because the square of any real number can never be negative.
Example 3.
F= {x:x, African countries who won the FIFA world cup till 2014} is an empty set, because no African country has ever won the FIFA world cup till 2014.
Example 4.
If A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {0, 2, 4, 6, 8, 10} then A\B={1,3,5,7,9}.
Example 5.
If universal set U is the set of all integers and A is the set of all negative integers, then (with respect to U), the complementary set of A will be A’ or Ac = {0,1,2,3,…}
Example 6.
If A = {a, b} and B = {b, c} , show that P (A) ∪ P(B) ⊆ P(A ∪ B).
Solution: Here, P(A) = { Ø }, {a},{b},{a,b}, P={ Ø } ,{b},{c },{ b,c}.
So, P (A) ∪ P(B) ⊆ P(A ∪ B).
Example 7.
The Venn diagram of A’, which is the complementary set of A with respect to the universal set U:
Example 8.
Two subsets of universal set U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are A=\ x / x prime number } and B={ x : x odd number}.
So A = {2, 3, 5, 7} and B = {1, 3, 5, 7, 9} .
Therefore, A cup B = {1, 2, 3, 5, 7, 9} ,A cap B= 3,5,7
A’ = {0, 1, 4, 6, 8, 9}, B’ = {0, 2, 4, 6, 8}
A^ prime cup B’ = {0, 1, 2, 4, 6, 8, 9} ,A^ prime cap B^ prime =0,4,6,8
(A cap B)^ prime = {0, 1, 2, 4, 6, 8, 9} ,(A cup B)^ prime =0,4,6,8.
Example 9.
If A=\ x / x positive integer} and number} then A and B are disjoint sets, A cap B= emptyset B=\ x / x negative real
Example 10.
If A = x:x in R and 0 <= x <= 2 and B = x:x in N and 0 <= x <= 2 then B ⊂ A,A cup B=A,A cap B = B = {1, 2} .
Example 11.
If A = x:x in R and 1 <= x <= 2 and B = x:x in R and 0 < x < 1 A cup B= x:x in Rand * 0 < x <= 2 and A cap B= ø ,So,A and B are disjoint sets.
Example 12.
A = {1, 2} B = {a, b, c} are two sets. So the Cartesian Product of these two sets is AB = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
Example 13.
Show that, A = {1,2,3,…,n} and B = {1,3,5,…, 2n-1} are equivalent sets, where n is a natural number.
Solution: A and B are equivalent, because the two sets have an one-one correspondence as shown below:
Remarks: The one-one correspondence indicated above can be described by A↔ B: k↔ 2k – 1, kЄ A.
Example 14.
Show that, the set of natural numbers N and the set of even natural numbers A = {2, 4, 6, 2n, }, are equivalent.
Solution: N = {1,2,3,…,n,…} and A are equivalent sets, because an one- one correspondence between the sets N and A can be established, which is shown below.
Remarks: The one-one correspondence indicated above, can be described by NA:n 2n, n Є N.
Example 15.
Out of 50 persons, 35 can speak English, 25 can speak both English and Bangla and every one can speak at least one of these two languages. How many persons can speak Bangla? How many persons can speak only Bangla?
Solution: Let, S be the set of the 50 persons, E be the set of persons among them who can speak in English, B be the set of persons who can speak in Bangla.
So according to the question, n(S) = 50 n(E) = 35 , n (E cap B)=25 and E cup B. Suppose, n(B) = x
Then from, n(S)=n(E cup B)= n(E) + n(B) -n(E cap B), we get 50 = 35 + x – 25 or, x = 50 – 35 + 25 = 40So n(B) = 40
therefore 40 persons can speak in Bangla.
Now, the set of those who can speak in only Bangla is ( B backslash E).
Let, n (B backslash E)=y.
As the sets E cap B and ( B backslash E) are disjoint and B =(E cap B) cup(B backslash E) [See Venn Diagram]
Example 16.
Each of the 35 girls in a class likes at least one activity among running, swimming and dancing. Among them 15 girls like running, 4 like all three of swimming, running and dancing, 2 like just running, 7 like both running and swimming but not dancing. z like both swimming and dancing but not running, 2x like only dancing, while 2 girls like only swimming.
1) Show these information in a Venn Diagram.
2) Find x.
3) Express the set of the girls who like both running and dancing, but not swimming.
4) What is the number of girls who like both running and dancing but not swimming?
Solution:
1) Suppose, set J = those who like running, S= those who like swimming, D= those who like dancing. Now let’s look at the Venn Diagram shown above.
2) From the Venn Diagram J’ = {the girls who do not like running}. So, n(J’) 35 15 20 or, 2x + x + 2 = 20 or, 3x = 18 or x = 6.
3) Set of the girls who like both running and dancing but not swimming: Jn Dns’.
4) n(JnDNS) = y is shown in the Venn Diagram and given that n(J) = 15. y + 4 + 7 + 2 = 15 or y = 2
Therefore, only 2 girls like running and dancing but not swimming.
Example 17.
Out of 24 students, 18 like to play basketball, 12 like to play volleyball. Given that U = the set of the students, B = the set of students who like to play basketball, V = the set of students who like to play volleyball. Suppose, n (B cap V)=x and explain the data below in Venn diagram:
1) Describe the set B cup V and express n (B cup V) in terms of x.
2) Find the possible minimum value of x.
3) Find the possible maximum value of x.
Solution:
1) B ∪ V is the set of the students who like to play basketball or vollyball
n(B cup V)= (18 – x) + x + (12 – x) = 30 – x
2) x or n (B cap V) is the smallest, when B cup V = U
So, n (B cup V)=n(U) or 30 – x = 24 or x = 6
.. The possible minimum value of x = 6
3) n (B cap V) is the largest when V subset B
Then, n (B cap V)=n(V) or x = 12
Possible maximum value of x = 12
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