1. This test does not include laws of indices, so \(x\times x \neq x^2\). Since we are dealing only with nouns \(-\times -\) does not simplify to \(1\). E.g. \(-x\times -x \neq x\times x \neq x^2\). This also means that \(\sqrt{x}\) is not considered to be equivalent to \(x^{\frac{1}{2}}\) under this test. In many situations this notation is taken mean the same thing, but internally in Maxima they are represented by different functions and not converted to a canonical form by the test. Extra re-write rules could be added to achieve this, which would change the equivalence classes.
1. This test does not include laws of indices, so \(x\times x \neq x^2\). Since we are dealing only with nouns \(-\times -\) does not simplify to \(1\). E.g. \(-x\times -x \neq x\times x \neq x^2\). This also means that \(\sqrt{x}\) is not considered to be equivalent to \(x^{\frac{1}{2}}\) under this test. In many situations this notation is taken mean the same thing, but internally in Maxima they are represented by different functions and not converted to a canonical form by the test. Extra re-write rules could be added to achieve this, which would change the equivalence classes.
2. By design, addition commutes with subtraction, so \( -1+2\equiv 2-1\) and multiplication commutes with division, so \( (ab)/c\equiv a(b/c) \).
2. By design, addition commutes with subtraction, so \( -1+2\equiv 2-1\) and multiplication commutes with division, so \( (ab)/c\equiv a(b/c) \).
3. By design \(-1/4x \neq x/4\) since we do not have the rule \( 1\times x \rightarrow x\). To establish this equivalence we would need a different answer test.
3. By design \(-1/4x \neq x/4\) since we do not have the rule \( 1\times x \rightarrow x\). To establish this equivalence we would need a different answer test.
4. This test can also be used to establish \(\{4,4\}\neq \{4\}\), but \(\{1,2\} = \{2,1\}\) since the arguments of the set constructor function are commutative. Sets are not associative, so \(\{1,2\}\neq \{\{1\},2\}\). (See Maxima's `flatten` command.)
4. This test can also be used to establish \(\{4,4\}\neq \{4\}\), but \(\{1,2\} = \{2,1\}\) since the arguments of the set constructor function are commutative. Sets are not associative, so \(\{1,2\}\neq \{\{1\},2\}\). (See Maxima's `flatten` command.)