Axiom Of Equality

Axiom of equality

(Reflexive axiom). Given any object \(x\), we have \(x=x\).
(Symmetry axiom) . Given any two object \(x\) and \(y\) of the same type, if \(x=y\), then \(y=x\).
(Transitive axiom). Given any three objects \(x,~y,~z\) of the same type, if \(x=y\) and \(y=z\), then \(x=z\).
(Substitution axiom). Given any two objects \(x\) and \(y\) of the same type, if \(x=y\), then \(f(x)=f(y)\) for all functions or operations \(f\).

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Exercise 3.3.1

\(=\) { \(\forall x\in X, f(x)=\tilde{f}(x)\),
\(~~~\) substitution axiom for any two objects of the same type }

\(\Rightarrow\) { \(X\), \(X'\) and \(X''\) are objects by Axiom 3.1,
\(~~~\) transitive axiom for any three objects of the same type }

\(\Rightarrow\) { \(X\) and \(X'\) are objects by Axiom 3.1,
\(~~~\) symmetry axiom for any two objects of the same type }

\(\Rightarrow\) { \(Y\), \(Y'\) and \(Y''\) are objects by Axiom 3.1,
\(~~~\) transitive axiom for any three objects of the same type }

\(\Rightarrow\) { \(f(x)\), \(g(x)\) and \(h(x)\) are objects by definition of function,
\(~~~\) transitive axiom for any three objects of the same type }

\(\Rightarrow\) { \(f(x)\) and \(g(x)\) are objects by definition of function,
\(~~~\) the symmetry axiom for any two objects of the same type }

\(\equiv\) { \(X\) is an object by Axiom 3.1,
\(~~~\) reflexive axiom for any object }

\(\equiv\) { \(X\) is an objects by Axiom 3.1,
\(~~~\) reflexive axiom for any object }

\(\equiv\) { \(Y\) is an object by Axiom 3.1,
\(~~\) reflexive axiom for any object }

\(\equiv\) { \(Y\) is an objects by Axiom 3.1,
\(~~~\) reflexive axiom for any object }

\(\equiv\) { \(f(x)\) is an object by definition of function,
\(~~~\) reflexive axiom for any object }

Substitution

Transitive

Reflexive

Symmetric
Exercise 2.2.4 (Propostion 2.2.13)

\(\Rightarrow\) { Substitution axiom: \(a=b\implies f(a)=f(b)\) }
Exercise 2.2.3 (Propostion 2.2.12)

\(\equiv\) { Substitution axiom: \(x=y\implies f(x)=f(y)\) }
Exercise 2.2.2 (Lemma 2.2.10)

\(\Rightarrow\) { Transitive axiom }

\(\equiv\) { Reflexive axiom }

\(\equiv\) { Symmetry axiom : \((c = a) \equiv (a = c)\) }
Exercise 2.2.1 (Proposition 2.2.5)

\(~~~~~\) Substitution Axiom: \(a = b \implies f(a) = f(b)\) }