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The concept of a logical whole in mathematics refers to a system where all parts are interconnected and interdependent, forming a complete and coherent entity. This concept is fundamental in various mathematical fields, such as set theory, algebra, calculus, and geometry.

1. Set Theory: In set theory, a logical whole is represented by a set, which is a collection of distinct objects, considered as an object in its own right. For instance, the set of all even numbers is a logical whole because it includes all numbers that satisfy the condition of being divisible by two. The concept of a logical whole in set theory is crucial for understanding mathematical structures and relationships.

2. Algebra: In algebra, a logical whole can be seen in the structure of equations and their solutions. For example, a linear equation represents a logical whole, where every part of the equation contributes to the solution. The coefficients, variables, and constants in the equation are all interconnected, and changing any part of the equation will affect the solution.

3. Calculus: In calculus, a logical whole is represented by a function and its derivatives or integrals. The function, its rate of change (derivative), and the area under its curve (integral) are all parts of a logical whole. They are interconnected and provide different perspectives on the same mathematical object.

4. Geometry: In geometry, a logical whole can be a geometric figure such as a triangle or a circle. All the properties of these figures (like sides, angles, radius, diameter) are interconnected. For instance, in a triangle, the sum of the angles is always 180 degrees, a fact that holds true due to the interconnectedness of the angles.

In conclusion, the concept of a logical whole is a fundamental aspect of mathematics. It highlights the interconnectedness and interdependence of different parts of a mathematical object or system. Understanding this concept can help in grasping the coherence and completeness of mathematical theories and structures.

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