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Question
Subtraction of integers is ___________________ .
Options
commutative but no associative
commutative and associative
associative but not commutative
neither commutative nor associative
Solution
neither commutative nor associative
Subtraction of integers is not commutative
For example: If a = 1 and b = 2, then both are integers
\[1 - 2 = - 1\]
\[2 - 1 = 1 \]
\[\Rightarrow - 1 \neq 1\]
\[\therefore a - b \neq b - a \forall a, b \in Z\]
Subtraction of integers is not associative.
For example: If a = 1, b = 2, c = 3, then all are integers
\[1 - \left( 2 - 3 \right) = 1 + 1\]
\[ = 2\]
\[\left( 1 - 2 \right) - 3 = - 1 - 3\]
\[ = - 4\]
\[ \Rightarrow 2 \neq - 4\]
\[ \therefore a - \left( b - c \right) \neq \left( a - b \right) - c , \forall a, b, c \in Z\]
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