Understanding Transitive Relations in Logic for a 74-Year-Old Student

To logically prove the conclusion that if A is bigger than B, and B is bigger than C, then A must be bigger than C, we can apply the principles of transitive relations. This concept is quite common in mathematics and logic.

Step 1: Define the Terms

First, let’s clarify what we mean by 'bigger.' When we say one object is bigger than another, we are comparing their sizes. In this case, we are using three objects A, B, and C, which follow these relationships:

• FACT #1: A is bigger than B (A > B)
• FACT #2: B is bigger than C (B > C)

Step 2: Understanding Transitive Property

In mathematics, there’s a property called the transitive property. It states that if A > B and B > C, then A must also be greater than C. This property can be written as:

If A > B and B > C, then A > C.

Step 3: Using Symbols to Prove the Conclusion

Let’s use symbols to clarify our relationships:

1. Since A is bigger than B, we write it as: A > B

2. Since B is bigger than C, we write it as: B > C

Using the transitive property:

From A > B and B > C, we can conclude that A > C.

Step 4: Visualizing the Concept

To visualize this, imagine three sticks of different lengths:

• Stick A: 6 inches
• Stick B: 4 inches
• Stick C: 2 inches

Here, you can easily see that:

• Stick A (6 inches) is longer than Stick B (4 inches).
• Stick B (4 inches) is longer than Stick C (2 inches).
• Therefore, Stick A is indeed longer than Stick C.

Step 5: Real-World Application

This logical reasoning isn’t just an exercise; it helps us understand relationships in various real-life scenarios. For example:

• When comparing age: If John is older than Mary, and Mary is older than Lisa, then John is older than Lisa.
• When comparing prices: If a pencil (A) costs more than a notebook (B), and the notebook (B) costs more than an eraser (C), then the pencil must cost more than the eraser.

In conclusion, using the transitive property, we have logically proven that if A is bigger than B, and B is bigger than C, then A is indeed bigger than C. This principle can be applied across various fields and is a fundamental concept in logic and mathematics.