Equality refers to the concept of two things being the same or equivalent in some way. In mathematics, equality means that two mathematical expressions have the same value. The axiom of extensionality is a fundamental principle in set theory that states that two sets are equal if and only if they have the same elements. This means that sets are equal if they contain the same elements, regardless of the order or repetition of those elements. The axiom of extensionality is crucial in defining and understanding the concept of equality in set theory. **

No, 112 is not an axiom. An axiom is a statement that is taken to be true without proof, and serves as a starting point for further reasoning. 112 is simply a number and does not fit the definition of an axiom. Axioms are typically statements about the fundamental properties of a mathematical system, such as the rules of logic or the properties of numbers, but 112 does not fit this criteria. **

Axiom 1 states that for any two distinct points, there exists exactly one line that passes through both points. For example, if we have two points A and B on a plane, there is only one line that can be drawn through both points. Axiom 5, also known as the Parallel Postulate, states that if a line intersects two other lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, will eventually intersect on that side. An example of this would be two parallel lines on a plane that are intersected by a third line, creating two interior angles that are less than 180 degrees, and the third line will eventually intersect the other two lines. **

This axiom means that if two things are equal to a third thing, then they are also equal to each other. In other words, if A = C and B = C, then A = B. This principle is fundamental in mathematics and logic, and it is often used in proofs and reasoning to establish relationships between different elements. It is also known as the transitive property of equality. **

An axiom is a statement or proposition that is considered to be self-evidently true and does not require proof. It is a fundamental principle that serves as a basis for reasoning or argumentation in a particular field of study. A simple example of an axiom is "A whole is greater than the sum of its parts." **

No, not every mathematical axiom is a dogma. Axioms are fundamental assumptions or principles that are accepted without proof in a particular system of mathematics. They serve as the foundation for mathematical reasoning and are subject to scrutiny and revision based on logical consistency and empirical evidence. Dogma, on the other hand, refers to a belief or principle that is considered to be unquestionably true and not subject to challenge or revision. While some axioms may be considered dogmatic in certain contexts, the nature of mathematical inquiry allows for the critical examination and potential revision of axioms based on logical and empirical considerations. **

The foundational axiom is a fundamental principle or assumption upon which a system of thought or belief is based. It serves as the starting point for reasoning and argumentation within a particular framework. In philosophy and mathematics, foundational axioms are used to establish the basic principles from which all other truths can be derived. These axioms are often self-evident or intuitively true, and they provide a solid foundation for building logical and coherent systems of knowledge. **

Newton's third axiom, which states that for every action, there is an equal and opposite reaction, works because it is based on the principle of conservation of momentum. When one object exerts a force on another object, the second object exerts an equal and opposite force back on the first object. This interaction allows for the conservation of momentum in a closed system, ensuring that the total momentum remains constant. This axiom has been consistently observed and tested in countless experiments, confirming its validity and effectiveness in describing the behavior of objects in motion. **