Question 1: Which of the following collections of objects are sets?

(i) All the months in a year

Answer: YES. It is a well-defined collection of distinct objects. There are months in a year. So, it is a definite set of elements.

(ii) All the rivers flowing in UP

Answer: YES. It is a well-defined collection of distinct objects.

(iii) All the planets in our solar system

Answer: YES. It is a well-defined collection of distinct objects.

(iv) All the interesting dramas written by Premchand

Answer: NO. It is not a well-defined collection of objects. Had it been “All dramas written by Premchand” then it could have been a set as that would be well defined.

(v) All the short boys in your class

Answer: NO. It is not a well-defined collection of objects. Had it been “All boys shorter than , then it would have become well defined. We need a benchmark to have a well-defined list of objects.

(vi) All the letters of the English Alphabets which precedes K

Answer: YES. It is a well-defined collection of distinct objects.

(vii) All the pet dogs in Nagpur

Answer: YES. It is a well-defined collection of distinct objects. Technically, all pets in Nagpur should be registered with a Government department. This might be a big list but still a well-defined list of objects.

(viii) All the dishonest shop owners in Noida

Answer: No. Not well defined.

(ix) All the students in your school with age exceeding 15 years

Answer: YES. It is a well-defined collection of distinct objects. School has a list of students that study there. The set would include all whose age is more than years.

(x) All the girls of Gita’s class who are taller than Gita

Answer: YES. It is a well-defined collection of distinct objects. We know all the girls studying in the class, we know Gita’s height and hence the set would contain distinct girls taller than Gita.

Question 2: Rewrite the following statements using the set notations:

(i) is an element of

Answer:

(ii) does not belong to set

Answer:

(iii) and are members of set

Answer:

(iv) and are equivalent sets

Answer:

(v) Cardinal number of set is

Answer:

Note: The number of distinct elements contained in a finite set is called the cardinal number of and is denoted by .

For example, if , then

(vi) is an empty set and is a non-empty set

Answer: and

Note: A set consisting of no elements is called an empty set or a null set or a void set. It is denoted by (called phai). We write

(vii) is a whole number, but is not a natural number

Answer: but

Note: Whole numbers : The numbers .

Natural numbers : The counting numbers , are called natural numbers

Question 3: Describe the following sets in roster form

(i)

Answer:

(ii)

Answer:

Note: You can calculate the factors using a tree method.

(iii)

Answer:

Note: and . This means is , and . Now calculate based on the given formula. Example: When .

(iv)

Answer:

Note: and . This means is . Now substitute the value of to calculate .

(v)

Answer:

Note: and , which means is . Now substitute the value of in the formula given for .

(vi)

Answer:

Note: If the first digit is , then the second digit is . So now try the values of as cannot be as it is a two digit number. Also by same logic, cannot be greater than either. Hence can only be and only.

(vii)

Answer:

Note: means that . Of these numbers needs to be divisible both by and .

(viii)

Answer:

Note: and which means that is . Now substitute the values.

(ix)

Answer:

Note: In careless, the alphabet and are repeated. We only need to take into account distinct elements only.

Question 4: Describe the following sets in set builder form

(i)

Answer:

(ii)

Answer:

(iii)

Answer:

(iv)

Answer: or

(v)

(vi)

Answer:

Note: If you notice, the numbers are square of .

(vii)

Answer:

Note: Integers (Z): Positive and negative counting numbers, as well as zero:

(viii) which is which means can be or

(ix)

Answer:

(x)

Answer:

(xi)

Answer:

(xii)

Answer:

(xiii)

Answer:

(xiv)

Answer:

Question 5: Separate finite and infinite sets from the following:

(i) Set of leaves on a tree

Answer: Finite. This is because in this case, the process of counting the leaves would surely come to an end.

(ii) Set of all counting numbers

Answer: Infinite. This is because there is no end to the numbers.

(iii)

Answer: Infinite. This is because there is no end to the numbers since .

(iv)

Answer: Finite. is whole numbers which are all natural numbers including . Since is less than , there are finite numbers to be counted.

(v)

Answer: Infinite. is an integer. Integers can be negative numbers too. So in this case, though is limited to less than on the positive scale, it can go to infinity on the negative scale.

(vi) Set of all triangles in a plane

Answer: Infinite. This is because there can be uncountable number of triangles in a plane.

(vii) Set of all points on a circumference of a circle

Answer: Infinite. This is because there can be uncountable number of points on a circumference of a circle.

(viii)

Answer: Infinite. This is because the set is not ending. It will keep having 1, 2, 3 repeated again and again

(ix)

Answer: Infinite.

Note: Rational numbers (Q): Numbers that can be expressed as a ratio of an integer to a non-zero integer. All integers are rational, but the converse is not true.

Question 6: Which of the following are empty sets?

(i)

Answer: Empty Set

Note: Natural Number . Since is given, it means, . But does not belong to and hence the set it empty or null.

(ii)

Answer: Empty Set

Note: Natural Number . Since is given, it means, . But does not belong to and hence the set it empty or null.

(iii)

Answer: Not an Empty Set

Note: Whole Number . Since is given, it means, . Therefore can be and hence it is not an Empty Set.

(iv)

Answer: Not an Empty Set

Note: Natural Number . Since , it means that can be . Hence is not an Empty Set.

(v)

Answer: Empty Set

Note: Natural Number . means that is which is not possible.

(vi)

Answer: Empty Set

Note: If , this means can be . All these numbers are divisible by a number. E.g. is divisible by by by and so on.

(vii)

Answer: Not an Empty Set.

Note: is an even prime number.

Question 7: Which of the following are pairs of equivalent sets?

Note: Two finite sets and are said to be equivalent, if , that is they have the same number of elements. An equivalent set is simply a set with an equal number of elements. The sets do not have to have the same exact elements, just the same number of elements.

(i) and

Answer: No

Note: and Therefore while

(ii) and

Answer: No

Note: and Therefore while

(iii) and

Answer: Yes

Note: and Therefore while Hence and are equivalent sets.

(iv) and

Answer: Yes

Note: and Therefore and Hence they are equivalent sets

Question 8: State whether the following statements are true or false.

(i) is not a set

Answer: False. It is a set. It contains elements.

(ii)

Answer: True. Both sets contain the same elements just in different order. The order of elements does not matter.

(iii) is a singleton set

Answer: True. The only element in the set would be . Hence it is a singleton set.

Note: a singleton, also known as a unit set, is a set with exactly one element. For example, the set is a singleton. The term is also used for a 1-tuple (a sequence with one element).

(iv)

Answer: True. The set is a null set as there is no element.

(v)

Answer: True. but is a natural number. Hence a null set.

(vi)

Answer: False. There is a valid element in the set which is .

(vii) If then

Answer: False.

(viii) If and , then

Answer: True. . Therefore . Therefore . Hence

(ix) If , then

Answer: False. Just because the number of elements in the sets and are equal, does not mean that the sets are also equivalent.

(x) If is a set of all consonants in English Alphabets, then

Answer: True

Note: The word consonant is also used to refer to a letter of an alphabet that denotes a consonant sound. The consonant letters in the English alphabet are , and usually and .

(xi)

Answer: False. Prime factors of are , and