Relaciones binarias
How to Calculate a Binary Relation
Introduction to Binary Relations
- Salvatore introduces the concept of binary relations, explaining that it involves determining a relationship between two sets, A and B.
- He presents the formula for calculating a binary relation, emphasizing that it must be included in the Cartesian product of the two sets.
Example Problem: Finding a Binary Relation
- The exercise begins with defining two sets: A = 1, 2, 3 and B = 1, 2. The goal is to find pairs (x,y) such that x + y < 4.
- Salvatore explains how to compute the Cartesian product of sets A and B by combining each element from set A with every element from set B.
Calculating Pairs in Cartesian Product
- He illustrates forming ordered pairs through combinations: (1,1), (1,2), (2,1), (2,2), (3,1), and (3,2).
- The total number of elements in the Cartesian product is calculated as 6 since there are three elements in set A and two in set B.
Establishing Valid Relationships
- Salvatore states that only certain pairs will satisfy the condition x + y < 4.
- He evaluates each pair against this condition; for example:
- Pair (1,1): 1+1 = 2 < 4 → valid.
- Pair (2,2): 2+2 = 4 → invalid.
Finalizing the Relationship Set
- After evaluating all pairs based on their sums:
- Valid pairs identified are (1,1), (1,2), (2,1).
- These valid pairs form the final binary relation which adheres to the specified condition.
Calculating Another Binary Relation
New Problem Statement
- Salvatore introduces another problem where he needs to calculate a relation given new sets A and B with specific conditions: x ≥ y.
Understanding Components of Sets
- Clarification is provided regarding which values can be taken by x and y based on their respective sets.
Direct Calculation Without Full Cartesian Product
- Instead of calculating the full Cartesian product again due to its complexity with larger sets:
- He directly assesses potential valid pairs based on their defined relationship.
Evaluating Each Pair Against Conditions
- For instance:
- Pair (3,1): Valid since 3 ≥ 1.
- Pair evaluations continue until all possible combinations are checked against x ≥ y criteria.
Conclusion on Valid Pairs
- Invalid combinations like (3,5): Not included as they do not meet criteria.
Understanding Binary Relations and Their Domains
Analyzing Relationships in Mathematics
- The discussion begins with analyzing the relationship between numbers, specifically focusing on the comparison of values such as 4 and 1, confirming that 4 is greater than or equal to 1.
- The speaker continues by comparing 4 with 2, establishing that this relationship holds true as well. They emphasize that certain comparisons do not hold, such as stating that 4.54 is less than 5.
Defining Domain and Range
- The concept of domain is introduced, defined as the set of first components from ordered pairs. In this case, the domain consists of the numbers 1, 3, 4.
- The range is discussed next; it includes second components from ordered pairs. Here, it identifies unique values like 1, 2, noting repetitions in data.
Relationship Between Domain and Range
- A connection between domain and range is highlighted: while the domain relates to first components only, the range can be a subset or part of a larger set.
- The speaker mentions calculating domains easily when given relationships and hints at using Cartesian products for further analysis without needing extensive notation.
Practical Example of Binary Relations
- An example involving real numbers is presented to illustrate how to calculate domains based on given conditions. This will be explored in more detail in future classes focused on binary relations within real numbers.