How Do Sets, Subsets, and Functions Work?

TL;DR

Cardinality counts the elements of a set, while subsets, power sets, and mappings describe relationships within and between sets. A function gives every domain element exactly one image, an injective function gives different inputs unique images, and a surjective function gives every codomain element a preimage. Domain and codomain choices therefore determine how a function is classified.

Transcript

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Key Insights

• Cardinality is the number of elements contained in a set. The lecture uses cardinal numbers to distinguish finite sets from infinite sets and notes that the natural numbers have infinite cardinality, while the real and complex numbers have uncountably many elements.
• A finite set is a set with finitely many elements or finite cardinality. An infinite set instead has infinitely many elements or infinite cardinality, making cardinality the basic measurement used in the lecture to separate these two types of sets.
• A proper subset is a subset that is not equal to the original set. The lecture illustrates that if a candidate subset equals set A, it is not proper, whereas smaller collections of elements from A can qualify as proper subsets.
• A power set is the collection of all subsets of a given set. For a finite set, the lecture states that the number of subsets is found by raising two to the number of elements in the original set.
• A function is a mapping in which every element of set A has exactly one image in set B. A domain element cannot have two images, and no element of the domain may be left without an assigned image.
• An injective function is also called a one-one function. Every element of set A must have a unique image in set B, and the lecture states that the cardinality of A must be less than or equal to the cardinality of B.
• A surjective function is also called an onto function. Every element of the codomain set B must have a preimage in domain set A, so no codomain element is left outside the range of the mapping.
• A bijective function is both one-one and onto. The lecture states that this requires the cardinality of set A to equal the cardinality of set B, while emphasizing that classification also depends on the selected domain and codomain.

Questions & Answers

Q: What is the cardinal number of a set?

The cardinal number, or cardinality, of a set is the number of elements contained in that set. The lecture uses cardinality to classify sets as finite or infinite. It states that the natural numbers have infinite cardinality and that the real and complex numbers contain uncountably many elements, a topic reserved for upcoming classes.

Q: What is the difference between finite and infinite sets?

A finite set has a finite number of elements, which means it has finite cardinality. An infinite set has infinitely many elements or infinite cardinality. The natural numbers are presented as an example with infinite cardinality, while the lecture also identifies the real and complex numbers as collections with uncountably many elements.

Q: What is a proper subset of a set?

A proper subset is a subset that does not equal the original set. If B is contained in A but B equals A, then B is not a proper subset of A. The lecture contrasts that case with smaller selections of elements from A, which can be proper subsets because they do not contain every element of A.

Q: How is the number of subsets of a finite set calculated?

The number of subsets of a finite set is calculated by raising two to the number of elements in that set, according to the lecture. The complete collection of these subsets is called the power set of A. This collection includes proper subsets as well as the subset that equals the original set.

Q: What conditions must a mapping satisfy to be a function?

A mapping is a function when every element of the domain, represented as set A, has exactly one image in the codomain, represented as set B. No domain element can have two different images, and no element of the domain can remain without an image. These conditions define the basic mapping discussed in the lecture.

Q: What is a one-one or injective function?

A one-one function, also called an injective function, assigns a unique image in set B to every element of set A. The lecture states that the cardinality of A must be less than or equal to the cardinality of B for such a function. Its classification also depends on the chosen domain and codomain.

Q: What is an onto or surjective function?

An onto function, also called a surjective function, is a mapping in which every element of the codomain set B has at least one preimage in the domain set A. No element of B is left without a corresponding input. The lecture states that A must have cardinality greater than or equal to B for an onto function.

Q: When is a function both one-one and onto?

A function is both one-one and onto when every domain element has a unique image and every codomain element has a preimage. The lecture states that the cardinality of set A equals the cardinality of set B in this case. Function classification still depends on how the domain and codomain are defined.

Summary & Key Takeaways

• Set theory begins with collections of elements and their cardinal numbers. A finite set has finite cardinality, while an infinite set has infinite cardinality. The natural numbers form an infinite set, and the lecture identifies the real and complex numbers as having uncountably many elements, with further discussion reserved for later classes.

• A subset contains elements drawn from another set, while a proper subset must not equal the original set. For a finite set, the stated method for finding its number of subsets is to raise two to the number of elements. The collection of all these subsets is called the power set.

• A function assigns every element of its domain exactly one image in its codomain. Injective functions give domain elements unique images, while surjective functions ensure every codomain element has a preimage. A function satisfying both conditions is one-one and onto, and the two sets then have equal cardinality.