Question

Let A= { x:x is a natural number <11 }
      B= { x:x is an even number and 1      C= { x:x is an integer and 15≤x≤25 }
(I) List the element of A,B,C
(ii) Find n(A),n(B),n(C).
(iii) State whether the following are True (T) or False (F)
      (a) 7∈B
      (b) 16 $ \notin $​A
      (c) {15,20,25}⊂C
      (d) {10,12}⊂B

Answer 1

Hint- In the question first find out the numbers according to given conditions on natural number, even number and on integer. Then use the concepts of set theory, to reach the solution of the problem.
Complete step-by-step solution -
In order to solve this problem, we must have good command on the Basic definition of Natural number and Integer and along with proper understanding of belongs to, set and subset that is being asked in the question.
(I) List the element of A, B, C

We Know That Natural numbers are a part of the number system which includes all the positive integers from 1 till infinity. It is an integer which is always greater than zero (0). Natural numbers are part of real numbers.

Let A= {x: x is a natural number <11}
This statements says that, x is a natural number which is less than 11 So according to definition of Natural number, possible value of x is equal to 1,2,3,4,5,6,7,8,9,10
And A consists of all possible values of x. In other words, A is a set of all values of x.
Hence A= {1,2,3,4,5,6,7,8,9,10},

We Know That, Any number that can be exactly divided by 2 is called an even number. Even numbers always end up with the last digit as 0, 2, 4, 6 or 8.
B= {x:x is an even number and 1This statement says that, x is an even number which is greater than 1 while less than 21. So according to definition of even number, possible value of x is equal to 2,4,6,8,10,12,14,16,18,20
And B consists of all possible values of x. In other words, B is a set of all values of x.
Hence B= {2,4,6,8,10,12,14,16,18,20}

We Know that an integer is a whole number (not a fraction) that can be positive, negative, or zero
C= {x:x is an integer and 15≤x≤25}
This statement says that x is an Integer which is greater than or equal to 15 while less than or equal to 25. So according to definition of Integer, possible value of x is equal to 15,16,17,18,19,20,21,22,23,24,25
And C consists of all possible values of x. In other words, C is a set of all values of x.
Hence C={15,16,17,18,19,20,21,22,23,24,25}


(ii) Find n(A), n(B), n(C).
n(A); This statements tells about the number of elements in set A
⇒A = {1,2,3,4,5,6,7,8,9,10}
In set A, there are total 10 elements So, n(A)=10

⇒B = {2,4,6,8,10,12,14,16,18,20}
In set B, there are total 10 elements So, n(B)=10

⇒C = {15,16,17,18,19,20,21,22,23,24,25}
In set C, there are total 11 elements So, n(C)=11

(iii) State whether the following are True (T) or False (F)

(a) 7∈B
This statements says that 7 belongs to set B
⇒B = {2,4,6,8,10,12,14,16,18,20}
By looking at set B, we get to know that Set B does not consist of 5.
Hence the statements 7 belongs to set B is false.
(b) 16 $ \notin $​A
This statements says that 16 does not belongs to set A
⇒A= {1,2,3,4,5,6,7,8,9,10}
By looking at set A, we get to know that Set A does not consist of 16.
Hence the statement 16 does not belong to set A is true.
(c) {15,20,25} ⊂C
This statement says that {15,20,25} is a proper subset of set C.
A proper subset ( $ \subset $ ) of a set S is a subset of S that is not equal to S.
In other words, if T is a proper subset of S, then all elements of T are in S but S contains at least one element that is not in T.
Here let Set T = {15,20,25} and set C which we already find equal to C = {15,16,17,18,19,20,21,22,23,24,25}
By looking at both set T and C, we get to know that all elements 15,20,25 of set T are present in set C and except (15,20,250), C also consist (16,17,18,19,21,22,23,24) which ensures that all elements of T and B are not same
Hence the statement {15,20,25} is a proper subset of set C.is true.

(d) {10,12} ⊂B
This statement says that {10,12} is a proper subset of set B.
Bu using above definition of proper subset
Here let Set T = {10,12} and set B which we already find equal to B = {2,4,6,8,10,12,14,16,18,20}
By looking at both set T and B, we get to know that all elements 10,12 of set T are present in set B and except (10,12), C also consist (2,4,6,8,14,16,18,20) which ensures that all elements of T and B are not same

Hence the statement {10,12} is a proper subset of set B is true.

Note- whenever we come up with this type of problems, We should have knowledge about the sets and get to know some basic definitions like subset, number of elements of a set, roster and set builder representation of a set. These are the key elements of this problem.