Introduction:
Set theory is a branch of mathematics dealing with the characteristics of well-defined collections of objects that may or may not be mathematical in nature, such as numbers or functions. The set theory is more significant as a foundation for accurate and adaptable terminology for the definition of complex and sophisticated mathematical concepts than as a direct application to everyday experience.
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The concept of a set is a fundamental aspect of modern mathematics. Today, practically every discipline of mathematics employs this concept. The ideas of relations and functions are defined using sets. The study of geometry, sequences, probability, and other subjects necessitates the understanding of sets. Georg Cantor, a German mathematician, invented the theory of sets (1845-1918). He originally came upon sets while working on "trigonometric series problems." We'll go over some basic set definitions and operations in this chapter.
A Set is an unordered collection of objects, known as elements or members of the set.
An element ‘a’ belonging to a set A can be written as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A.
NCERT Notes Subject Wise Link:
The concept of a set is a fundamental aspect of modern mathematics. Today, practically every discipline of mathematics employs this concept. The ideas of relations and functions are defined using sets. The study of geometry, sequences, probability, and other subjects necessitates the understanding of sets. Georg Cantor, a German mathematician, invented the theory of sets (1845-1918). He originally came upon sets while working on "trigonometric series problems." We'll go over some basic set definitions and operations in this chapter.
Although, in the JEE test, there is just one question from Sets.
However, it is still important.
Set theory is as a topic is not very important but when its use comes in functions and relations then it becomes a very important and basic concept.
NCERT Solutions Subject wise link:
NCERT Exemplar Solutions Subject wise link:
Set theory is the mathematical study of well-defined collections of objects known as members, or elements, of the set.
A proper subset is one that contains a few elements of the original set whereas an improper subset, contains every element of the original set along with the null set. For example, if set A = {2, 4, 6}, then, Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or {}.
If a set does not contain any element or member then the set is called a null set. Null set is also called a void set or empty set. The symbol used to represent an empty set is – ϕ ,{}. Examples: 1) Let A = {x :1 < x < 2, x is an integer } be a null set because there is no integer between numbers 1 and 2.
The complement of a set is the set that contains all of the components of the universal set that are not present in the provided set.
In mathematics, a singleton, also known as a unit set, is a set with exactly one element. For example, the set {null } is a singleton containing the element null.
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Correct Answer: presence of mammary gland, sweat glands and diaphragm
Solution : The Correct Answer is presence of mammary gland, sweat glands and diaphragm
Mammals are distinguished by their placenta, hairy skin, mammary glands, muscular diaphragms, and the ability to give birth to young. Aves have four chambers in their hearts and worm blood. The reptile class crocodiles also have a four-chambered heart. Because of their body hair, mammals can be recognized. Among all living things, mammals rank among the most intelligent. A wide range of animals, including cats, people, and whales, are considered mammals.
Correct Answer: (142, 83, 69)
Solution : Given:
(168, 35, 143); (182, 65, 127)
In the given sets, subtract the third number from the first number and then add 10.
(168, 35, 143)→168 – 143 = 25; 25 + 10 = 35
(182, 65, 127)→182 – 127 = 55; 55 + 10 = 65
Let's check the options –
First option: (142, 83, 69)→142 – 69 = 73; 73 + 10 = 83
Second option: (253, 99, 154)→253 – 154 = 99; 99 + 10 = 109 ≠ 99
Third option: (203, 89, 117)→203 – 117 = 86; 86 + 10 = 96 ≠ 89
Fourth option: (159, 62, 87)→159 – 87 = 72; 72 + 10 = 82 ≠ 62
So, only the first option follows the same pattern as followed by the given set of numbers. Hence, the first option is correct.
Correct Answer: (19, 13, 4)
Solution : Given:
(15, 9, 4); (7, 2, 3)
Here, (15, 9, 4)→(9 + 4) + 2 = 13 + 2 = 15
(7, 2, 3)→(2 + 3) + 2 = 5 + 2 = 7
Let's check the options –
First option: (21, 12, 6)→(12 + 6) + 2 = 18 + 2 = 20 ≠ 21
Second option: (19, 13, 4)→(13 + 4) + 2 = 17 + 2 = 19
Third option: (24, 9, 11)→(9 + 11) + 2 = 20 + 2 = 22 ≠ 24
Fourth option: (18, 7, 7)→(7 + 7) + 2 = 14 + 2 = 16 ≠ 18
So, only the second option follows the same pattern as followed by the given set of numbers. Hence, the second option is correct.
Correct Answer: (2, 7, 28)
Solution : Given:
(3, 4, 24); (4, 5, 40)
In the above-given sets, multiply 2 by the product of the first and second numbers.
(3, 4, 24)→3 × 4 = 12; 12 × 2 = 24
(4, 5, 40)→4 × 5 = 20; 20 × 2 = 40
Let's check each option –
First option: (4, 6, 35)→4 × 6 = 24; 24 × 2 = 48 ≠ 35
Second option:(5, 7, 42)→5 × 7 = 35; 35 × 2 = 70 ≠ 42
Third option: (3, 7, 21)→3 × 7 = 21; 21 × 2 = 42 ≠ 21
Fourth option: (2, 7, 28)→2 × 7 = 14; 14 × 2 = 28
So, (2, 7, 28) follows the same pattern. Hence, the fourth option is correct.
Correct Answer: (13, 19, 64)
Solution : Given:
(13, 14, 54); (15, 18, 66)
Like, (13, 14, 54)→(13 + 14) × 2 = 27 × 2 = 54
(15, 18, 66)→(15 + 18) × 2 = 33 × 2 = 66
Let's check the options –
First option: (11, 13, 52)→(11 + 13) × 2 = 24 × 2 = 48 ≠ 52
Second option: (15, 17, 66)→(15 + 17) × 2 = 32 × 2 = 64 ≠ 66
Third option: (13, 19, 64)→(13 + 19) × 2 = 32 × 2 = 64
Fourth option: (12, 34, 90)→(12 + 34) × 2 = 46 × 2 = 92 ≠ 90
So, only the third option follows the same pattern as followed by the given set of numbers. Hence, the third option is correct.