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A short class with Sarah and her students. The other ways are under More.
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Today we're going to start with the definition of a relation. Can anyone tell me what a relation actually is?
Isn't it something to do with how two things connect?
Exactly! A relation between two sets A and B is a subset of the Cartesian product A×B. In simpler words, it consists of ordered pairs where the first element is from A and the second is from B. For example, if we have A = {1, 2, 3} and B = {a, b, c}, one possible relation could be R = {(1, a), (2, b), (3, c)}.
Wait, so it’s like matching pairs?
Yes! You can think of it as matching pairs from two different groups. Now, can anyone tell me what a Cartesian product is?
Isn't that when you combine all elements from both sets in ordered pairs?
Spot on! The Cartesian product A×B creates all possible pairs from the two sets. Let’s move on to discuss different types of relations.
Now that we understand what a relation is, let’s break down the types of relations. First, we have the reflexive relation. Can anyone guess what that means?
Does it involve pairs like (x, x)?
Exactly! A relation R is reflexive if every element x in set A also has the pair (x, x) in R. For example, if A = {1, 2, 3}, then the reflexive relation could be R = {(1, 1), (2, 2), (3, 3)}. What about symmetric relations?
That sounds like if (a, b) is in R, then (b, a) is also in R?
Precisely! Next, we have transitive relations. Can anyone explain what that means?
If you have (a, b) in R and (b, c) in R, then (a, c) should also be in R?
Yes, wonderful! And remember, an anti-symmetric relation happens when if (a, b) and (b, a) are both present, then a must equal b. Learning these types helps in identifying how relations behave.
Let's switch gears to functions. Can anyone tell me what makes a function different from a relation?
Functions map inputs to one specific output, right?
Exactly! A function is a special kind of relation where each input from the domain maps to exactly one output in the co-domain. We denote this as f: A → B. For instance, if A = {1, 2, 3} and B = {a, b, c}, then a function could look like f = {(1, a), (2, b), (3, c)}.
What about injective or surjective? I’ve heard those terms.
Great question! An injective function means different inputs yield different outputs. Surjective, or onto, means every element in the co-domain is mapped by at least one element in the domain. And if a function is both, we call it bijective. Let’s take an example: if f = {(1, a), (2, b), (3, b)}, it’s surjective because both 2 and 3 map to b, while f = {(1, a), (2, b), (3, c)} is bijective. This is crucial as we delve deeper into functions later.
Now let's discuss the composition of functions. If we have two functions f and g, how do we define their composition?
Isn’t it like applying one function after another? Like, g(f(x))?
Correct! The composition of f and g, denoted g∘f, is defined as g(f(x)) for all x in A. This means you apply function f first, then use the result in function g. Can anyone tell me what the inverse of a function is?
Does it reverse the operation of f?
Exactly! The inverse f⁻¹ undoes what f does. A function has an inverse only if it is bijective. So, if we were to say f: A → B, the inverse would be f⁻¹: B → A. This concept is vital to understand how functions relate to each other.
A relation between two sets A and B is a subset of A×B, meaning it consists of ordered pairs where the first element comes from A and the second from B. An example is R = {(1, a), (2, b), (3, c)} if A = {1, 2, 3} and B = {a, b, c}.
Types of Relations:
For every x in A, the pair (x, x) belongs to R.
If (a, b) is in R, (b, a) must also be in R.
If (a, b) and (b, c) are in R, then (a, c) must be in R.
If (a, b) and (b, a) are in R, then a must equal b.
A relation that is reflexive, symmetric, and transitive.
A function is a specific type of relation where each element in the domain (set A) corresponds to exactly one element in the co-domain (set B). Formally, f: A → B.
Types of Functions:
Different elements in the domain map to different elements in the co-domain.
Every element in the co-domain has at least one element from the domain mapping to it.
A function that is both injective and surjective.
Domain, co-domain, and range are discussed to understand how functions operate.
These operations are essential for the manipulation and understanding of functions in various contexts.
By mastering these definitions and properties, students gain tools to explore deeper mathematical currents and applications.
Example of a Relation: R = {(1, a), (2, b), (3, c)} for sets A = {1, 2, 3} and B = {a, b, c}.
Reflexive Relation Example: R = {(1, 1), (2, 2), (3, 3)} for set A = {1, 2, 3}.
Symmetric Relation Example: R = {(1, 2), (2, 1)} where both pairs are included.
Transitive Relation Example: R = {(1, 2), (2, 3), (1, 3)} showing linkage.
Injective Function Example: f = {(1, a), (2, b), (3, c)}; distinct mappings.
Surjective Function Example: f = {(1, a), (2, b), (3, b)}; b is covered by multiple inputs.
Bijective Function Example: f = {(1, a), (2, b), (3, c)}; one-to-one correspondence.
In a relation, pairs we find, ordered kindly, two sets aligned.
Once upon a time, in the land of Sets, A met B to form Relations. They decided to pair up based on their unique identities, creating Functions that held secrets of their own. Together, they composed stories and sometimes even reversed their tales.
RATS: Reflexive, Anti-symmetric, Transitive relations help us remember key types.
FIR: Function Is Royal, remember it maps inputs to unique outputs.
Term
What is a reflexive relation?
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What defines a bijective function?
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What is the composition of functions?
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Define surjective function.
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A subset of the Cartesian product of two sets, consisting of ordered pairs.
A special type of relation where each input is related to exactly one output.
A relation where every element has itself as a pair.
A relation where if (a, b) is in R, then (b, a) is also in R.
A relation where if (a, b) and (b, c) are in R, (a, c) must also be in R.
A relation where if (a, b) and (b, a) are in R, then a must equal b.
A relation that is reflexive, symmetric, and transitive.
A function where different inputs map to different outputs.
A function where every output in the co-domain has at least one input in the domain.
A function that is both injective and surjective.
The set of all possible input values for a function.
The set of possible output values for a function.
The set of actual output values that a function takes.
The process of applying one function to the results of another.
A function that reverses the effect of another function.
The section provides foundational definitions and concepts related to relations and functions, crucial for understanding complex mathematical structures.
This section explores the definitions of relations and functions, including types of relations (reflexive, symmetric, transitive, anti-symmetric, and equivalence relations) and classes of functions (injective, surjective, bijective). Each concept is explained with examples and key terminology.
In this section, we delve into the fundamental definitions surrounding relations and functions, two core concepts in mathematics that underpin the study of more advanced topics such as calculus and algebra. Understanding relations is crucial as they describe the connections between elements of different sets. A relation is defined as a subset of the Cartesian product of two sets, represented by ordered pairs.
By mastering these definitions and properties, students gain tools to explore deeper mathematical currents and applications.