Question

Let * be a binary operation on N given by a * b = ab. Find 20 * 16 and also find the identity element of * in N.

Answer 1

Hint: A binary operation is a calculation that combines two elements to produce a new one. To find the identity element of the * operation use the property a * e = e * a = a.
Complete step by step answer:
If a set X contains elements and an operator is defined on the set such that it takes two elements from the set and returns another element that also belongs to the set is called a binary operation.
The binary operation * is defined on N and is given as follows:
\[a*b = ab...........(1)\]
We need to find the operation 20 * 16, using equation (1), we have:
\[20*16 = 20 \times 16\]
Simplifying, we have:
\[20*16 = 320\]
Now, we need to find the identity element of the * operator.
The identity element e when operated on any element a in N, it gives back the same number. Hence, we have:
\[a*e = e*a = a\]
Now, equating the first and the third term, we have:
\[a*e = a\]
Using equation (1), we have:
\[ae = a\]
Dividing both sides by a, we have:
\[e = 1\]
Now, we equate the second term and the third term to check if 1 is the identity element.
\[e*a = a\]
Using equation (1), we have:
\[ea = a\]
Dividing both sides by a, we have:
\[e = 1\]
Hence, the identity element of the * operator is 1.

Note: For finding the identity element you need to verify both the conditions \[e*a = a\] and \[a*e = a\], and if and only if both conditions give the same identity element, the identity exists, otherwise it does not exist.