The Axioms of Equality

In the previous post, we have discussed about the axioms of real numbers. In this post, we learn about five more axioms, the axioms of equality in real numbers \mathbb{R}. Again, most of you are already familiar with these axioms since they are quite intuitive. We will emphasize these axioms because we are going to use them in proving theorems later.

Axiom E1: Relfexivity of Equality

For any a in \mathbb{R}, a = a.

Axiom E2: Symmetry for Equality

For any a and b in \mathbb{R}, if a = b, then b = a.

Axiom E3: Transitivity of Equality

For any a, b, and c in \mathbb{R}, if a = b and b = c, then a = c.

Axiom E4: Addition Property of Equality

For any a, b, and c in \mathbb{R}, if a = b, then a + c = b + c.

Axiom E5: Multiplication Property of Equality

For any a, b, and c in \mathbb{R}, if  a = b, then ac = bc.

 

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