Question

State and prove cancellation laws on groups.

Answer 1

Hint: Here we will first assume the group and then we will assume the elements of that group. We will then state the cancellation laws on groups. We will then prove both right cancellation laws and left cancellation laws using associative law of multiplication.

Complete step-by-step answer:
Let \[G\] be the group and \[a\], \[b\] and \[c\] be the elements i.e. \[a,b,c \in G\].
So, according to the cancellation laws,
If \[a \cdot b = a \cdot c\] or \[b \cdot a = c \cdot a\] then \[b = c\].
Now, we will first prove the left cancellation laws on groups.
For that, we will first consider the given mathematical expression.
\[a \cdot b = a \cdot c\]
On pre multiplying by the term \[{a^{ - 1}}\] on both sides, we get
\[ \Rightarrow {a^{ - 1}} \cdot \left( {a \cdot b} \right) = {a^{ - 1}} \cdot \left( {a \cdot c} \right)\]
On applying associative law, we get
\[ \Rightarrow \left( {{a^{ - 1}} \cdot a} \right) \cdot b = \left( {{a^{ - 1}} \cdot a} \right) \cdot c\]
Let \[{a^{ - 1}} \cdot a = e\]
Now, substituting \[{a^{ - 1}} \cdot a = e\] in the above equaton, we get
\[ \Rightarrow e \cdot b = e \cdot c\]
On further simplifying the terms, we get
\[ \Rightarrow b = c\]
Hence, this is the required proof of left cancellation laws.
Now, we will prove the right cancellation laws.
For that, we will first consider the given mathematical expression.
\[b \cdot a = c \cdot a\]
On pre multiplying by the term \[{a^{ - 1}}\] on both sides, we get
\[ \Rightarrow \left( {b \cdot a} \right) \cdot {a^{ - 1}} = \left( {c \cdot a} \right) \cdot {a^{ - 1}}\]
On applying associative law, we get
\[ \Rightarrow b \cdot \left( {a \cdot {a^{ - 1}}} \right) = c \cdot \left( {a \cdot {a^{ - 1}}} \right)\]
Let \[a \cdot {a^{ - 1}} = e\]
Now, we will substitute the value here
\[ \Rightarrow b \cdot e = c \cdot e\]
On further simplifying the terms, we get
\[ \Rightarrow b = c\]
Hence, this is the required proof of right cancellation laws.

Note: Here we have used the associative law of multiplication. The associative property of multiplication is defined as a math rule that states that the way in which factors are grouped in a multiplication problem in such a way that it does not change the product. We should not get confused between associative and commutative law of multiplication. Commutative law of multiplication states that the order of the numbers doesn’t affect the answer. Commutative and associative law is only applicable while addition and multiplication of terms.