Real Analysis | Set Theory | Set Theory Basic Definition & Examples

Summary

In this video, Dr. Gajendra Purohit introduces viewers to the fundamentals of real analysis, beginning with set theory. He explains key concepts such as finite and infinite sets, cardinality, subsets, and power sets, gradually progressing to functions, including one-one (injective) and onto (surjective) functions, with practical examples and graphical interpretations. The lecture is designed for students preparing for competitive exams like IIT JAM, CSIR NET, GATE, and BSc courses. Dr. Purohit also highlights his YouTube channels and playlists, promising more advanced topics, theorems, and proofs in upcoming videos to strengthen higher mathematics understanding.

Takeaways

• 😀 Dr. Gajendra Purohit provides video lessons on Engineering Mathematics, BSc, IIT JAM, CSIR NET, and GATE preparation.
• 😀 The video series starts with the basics of Real Analysis, beginning with Set Theory.
• 😀 Cardinality refers to the number of elements in a set; finite sets have a limited number of elements, while infinite sets have uncountable elements.
• 😀 The cardinality of natural numbers is infinite, whereas real and complex numbers have uncountable elements.
• 😀 A subset B of a set A can be a proper subset if B is not equal to A; the total number of subsets of a set is called the power set.
• 😀 In a function, each element of the domain (set A) must have a unique image in the codomain (set B), with no elements left without an image.
• 😀 A one-one function (injective) maps each element of set A to a unique element in set B; an onto function (surjective) ensures every element of set B has a pre-image in set A.
• 😀 The function can be both one-one and onto (bijective) when the cardinality of set A equals the cardinality of set B.
• 😀 Graphical interpretation helps understand onto functions: a line parallel to the x-axis must intersect the graph at least once for the function to be onto.
• 😀 Future videos will cover theorems with proofs, and additional playlists are available for topics like infinite series and higher mathematics.
• 😀 The channel encourages viewers to like, share, subscribe, and press the bell icon to stay updated on new content.

Q & A

What is the cardinality of a set and how is it denoted?

Cardinality refers to the number of elements in a set. For finite sets, it is the actual number of elements, while for infinite sets, it can be countable (like natural numbers) or uncountable (like real and complex numbers).

What is the difference between finite and infinite sets?

A finite set has a limited number of elements, whereas an infinite set has unlimited elements. Examples include natural numbers (infinite) and a set of letters in the English alphabet (finite).

What is a proper subset and how does it differ from a regular subset?

A proper subset of a set A is a subset B where B contains some but not all elements of A, meaning B ≠ A. A regular subset can include the entire set itself.

How do you calculate the number of subsets of a set?

The number of subsets of a set with n elements is given by 2^n, which is also called the power set of the set.

What are the conditions for a function to be one-one (injective)?

A function is one-one if every element of set A maps to a unique element of set B, meaning no two elements in A share the same image. The cardinality of A must be less than or equal to B.

What are the conditions for a function to be onto (surjective)?

A function is onto if every element of set B has at least one pre-image in set A, meaning no element in B is left unmapped. The cardinality of A must be greater than or equal to B.

What does it mean for a function to be both one-one and onto (bijective)?

A bijective function is both injective and surjective, meaning every element of A maps uniquely to B and every element of B has a pre-image in A. This requires the cardinality of A and B to be equal.

How can you visually determine if a function is onto using a graph?

A function is onto if any horizontal line drawn across the graph intersects it at least once. This ensures every value in the codomain has a pre-image in the domain.

What are the other names for one-one and onto functions?

A one-one function is also called injective, and an onto function is called surjective.

Why is it important to understand domain and codomain when analyzing functions?

The domain and codomain determine the mapping rules of a function. Whether a function is one-one, onto, or bijective depends on the relationship between elements in the domain and codomain, including their cardinalities.

Can you give an example of a function that is one-one but not onto?

If set A has 3 elements and set B has 4 elements, a function can map each element of A to a unique element of B, making it one-one. However, since one element of B will remain unmapped, it is not onto.

Can you give an example of a function that is onto but not one-one?

If set A has 4 elements and set B has 3 elements, a function can cover all elements of B (making it onto), but since A has more elements than B, some elements of A must share the same image, so it is not one-one.