memahami korespondensi satu satu kelas 8

Summary

In this educational video, the concept of 'one-to-one correspondence' in mathematics is thoroughly explained. The host introduces the topic by defining it as a relation where each element from one set (domain) is paired with exactly one element from another set (codomain), and vice versa. Through various examples and visual aids like diagrams, the video demonstrates how to identify and calculate one-to-one correspondences. Additionally, viewers learn to calculate the number of possible one-to-one correspondences using a factorial formula. The video concludes with practice problems to reinforce understanding.

Takeaways

• 😀 One-to-one correspondence (korespondensi satu-satu) is a relation that matches each element of one set (domain) with exactly one element of another set (codomain), and vice versa.
• 😀 A valid one-to-one correspondence means every member of the domain has a unique pair in the codomain, and every member of the codomain has a unique pair in the domain.
• 😀 For example, if set A has elements {1, 2, 3} and set B has elements {A, B, C}, the relation is one-to-one if 1 is paired with A, 2 with B, and 3 with C.
• 😀 A relation from set A to set B is not one-to-one if any domain element is paired with more than one codomain element or if any codomain element is not paired with any domain element.
• 😀 The number of possible one-to-one correspondences between two sets of equal size can be calculated using factorial notation (n!).
• 😀 For sets with 3 elements each, the number of one-to-one correspondences is 3! = 6.
• 😀 To calculate the number of one-to-one correspondences between two sets, multiply the size of the sets and apply the factorial formula, e.g., n! = n * (n-1) * ... * 2 * 1.
• 😀 If set P has 6 elements and set Q has 6 elements, the total number of one-to-one correspondences between them is 6! = 720.
• 😀 In practice, several different one-to-one correspondences can be formed between sets, such as pairing the first element of P with different elements in Q.
• 😀 A specific example of three different one-to-one correspondences between two sets of six elements each includes pairing 1 with A, 2 with B, 3 with C, etc., in various orders.

Q & A

What is the definition of one-to-one correspondence?

One-to-one correspondence is a relation that pairs each element of a set (domain) with exactly one element of another set (codomain), and vice versa. Every element in the domain must have one unique partner in the codomain, and every element in the codomain must have one unique partner in the domain.

How are the sets in a one-to-one correspondence represented in the script?

In the script, the sets are represented as set A (with elements 1, 2, 3) and set B (with elements A, B, C). The relationship between the sets is described as a one-to-one correspondence when each element from set A is paired with exactly one element from set B.

What are the two main conditions for a one-to-one correspondence to exist?

The two main conditions are: (1) Each element in the domain must have exactly one corresponding element in the codomain, and (2) each element in the codomain must have exactly one corresponding element in the domain.

What happens if one of these conditions is violated?

If either condition is violated (i.e., an element in the domain has more than one partner, or an element in the codomain has no partner), the relation is no longer a one-to-one correspondence.

What is the formula to calculate the number of one-to-one correspondences between two sets?

The formula to calculate the number of one-to-one correspondences is the factorial of the number of elements in the set, denoted as n!. For example, if there are 3 elements in each set, the number of one-to-one correspondences is 3! = 6.

How is the factorial of a number calculated?

The factorial of a number n (denoted as n!) is calculated by multiplying all positive integers less than or equal to n. For example, 3! = 3 * 2 * 1 = 6.

What is the significance of the sets having the same number of elements for a one-to-one correspondence?

For a one-to-one correspondence to be possible, the two sets must have the same number of elements. This ensures that each element in one set can be paired uniquely with an element in the other set.

In the example with sets P and Q, how many one-to-one correspondences can be formed?

In the example, both set P and set Q have 6 elements each. The number of one-to-one correspondences that can be formed between these sets is 6!, which equals 720.

How does the script illustrate different one-to-one correspondences with the sets?

The script uses diagrams and examples to show different one-to-one correspondences. It illustrates various pairings between elements of set A and set B, ensuring that each element from the domain and codomain is paired exactly once, with 6 possible unique pairings in total.

How are three examples of one-to-one correspondences between set P and set Q presented in the script?

The script presents three examples of one-to-one correspondences as follows: 1) (1, a), (2, b), (3, c), etc.; 2) (1, b), (2, c), (3, d), etc.; and 3) (1, c), (2, d), (3, e), etc. These examples show different possible pairings that maintain the one-to-one correspondence.