2.15 Algebraic Structure:
We are aware of addition,
subtraction, multiplication and division operations. They can be termed as
basic operations.
Can we have more
operations? Yes, for example we can have operations like:
1
Sum of two numbers divided by 2 { (a+b)/2}
– Average
2. A number raised to the power of another number (m^{n})
– Exponentiation
Recall:
Expression 
Short form(Symbol) 
Belongs to 
_{} 
Does not belong to 
_{} 
For Every 
_{} 
There exists 
_{} 
Such that 
: 
Set
of Natural numbers N = {1,2,3,4 …} = { n: n _{} natural numbers}
Set of whole numbers W = {0,1,2,3,….} = {n: n =0, and
n_{}{N}}
2.15 Example1:
S = {2, 4, 8,16….} = { Numbers raised to the powers of 2 starting with
2} = {2^{m} ;where m is any
natural number >1}
Let us do addition,
multiplication, exponentiation operations on this set S. and we observe
1. The sum of any two
numbers of S does not belong to S (ex; 6(=2+4),10(=2+8),12(=4+8)
do not belong to S)
2. The product of any two numbers of S belongs to the set S. Why?
(If 2^{m} and 2^{n}
are two
numbers then the product is2^{m+n
}[= (2^{m} )*(2^{n})] which belongs to S)
3. The exponentiation of any 2 numbers of S
belongs to the set S . Why ?
(If 2^{m} and 2^{n}
are two
numbers then the exponent 2^{mz} [=(2^{m} )^{z }where z =2^{n
} belongs to S)
Observation: The result of sum ‘operation’ on any element of S does not belong to S. But the
result of multiplication and
exponentiation ‘operations’ on any element of S belongs to S
Definition:
1.If
a, b _{} A and the result of ‘operation’ on a, b _{}A, then we say
that A satisfies closure property w.r.t ‘operation’ ( w.r.t is short form
for with respect to)
2. If a, b _{} A and c = a‘operation’b _{}A, then we say that ‘operation’ is
a Binary opeartion. By convention this ‘operation’
is denoted by ‘*’(not to be confused with
multiplication)and
a
‘operation’ b is read as ‘a star b’.
In the case of example 1,
we note that S does not satisfy closure property w.r.t
addition and hence the addition ‘operation’ is
not a Binary operation on S.
But multiplication and
exponentiation ‘operations’ satisfy closure
property and these two ‘operations’ are binary
operations on S
Examples:
No. 
Set 
Star (_{}) 
Observations 
Conclusion 
Reasoning

1 
N =
{1,2,3; Natural Numbers} 
sum 
_{},a,b _{}N, a+b _{}N 
N
Satisfies Closure property w.r.t + 
Sum of
2 natural numbers is a natural number 
2 
N = {1,2,3;
Natural Numbers} 
product 
_{},a,b _{}N, a* b _{}N 
N
Satisfies Closure property w.r.t * 
Product of 2 natural numbers is a natural number 
3 
A =
{1,3,5: Odd numbers} 
sum 
_{},a,b _{}N, a+b _{}N 
A does not satisfy closure property w.r.t
+ 
Sum of
2 odd number is an even number 
4 
B =
{1,3,5: Odd numbers} 
product 
_{},a,b _{}N, a*b _{}N 
B
satisfies closure property w.r.t * 
Product of 2 odd number is an odd number 
5 
Z
=(0,1,1,2,2: Integers) 
Average 
_{},a,b _{}Z, a_{}b=(a+b)/2 _{}Z 
Z does not satisfy closure property w.r.t the ‘operation:
Average’ 
Operation
0 star 1 = (0+1)/2 which is a fraction and not an integer 
6 
Q =
(p/q, where p,q _{}Z and q _{}0 
Division

_{},a,b _{}Q, a/b _{}Q, though 0 _{} Q 
Q does not Satisfy closure property w.r.t the ‘operation: /’ 
Division
of rational number by 0 is undefined ( Though
0_{} Q, 1/0 _{}Q) 
Relationship between Closure
property and Binary operation:
If any set satisfies the closure property w.r.t an operation then that operation is a binary
operation and conversely if an operation on a set is binary operation then the
set satisfies closure property w. r. t that operation.
Definition:
The ‘algebraic structure’ is a pattern such that the
non empty set S and the ‘operation(*)’ on S is a binary operation. The algebraic
structure is denoted by the expression (S,*)
In the examples listed
above we can see that
(N,+),(N,*),(B,*) are all
Algebraic structures and (A,+),(Z, average), (Q,/) are not Algebraic structures.
2.15 Summary of learning
No 
Points to remember 
1 
If a, b _{} A and the result of
‘operation’ on a, b _{}A, then we say
that A satisfies closure property w.r.t ‘operation’ 
2 
If any set
satisfies the closure property w.r.t an operation then
that operation is a binary operation and conversely if an operation on a set
is binary operation then the set satisfies closure property w. r. t that
operation. 
3 
The Algebraic structure is a pattern such that the non
empty set S and the ‘operation(*)’ on
S is a binary operation and is denoted by the expression (S,*) 