A binary operation is a function f (x, y) that is applied to two members of the same set S to produce a result that is also an element of the set S. Addition of integers, multiplication of whole numbers, and other binary operations are examples. A binary operation is a rule that is applied to two items of a set, with the resultant element belonging to the same set as the original.
If S is a non-empty set and * is a binary operation on S, it should satisfy the condition that says, if a ∈ S and b S, then a * b S, a, b S. I according to the definition of binary operations.
Figure 1
Binary operators are categorized as mathematical, logical, or relational. operations.
1. Addition (+)
2. Subtraction (-)
3. Multiplication (*)
4. Division (/)
These are the four basic mathematical operations.
The following are the properties of binary operations:
Commutative property: If a×b = b × a, for any (a, b) A, a binary operation * on that set is commutative (non-empty set). For a = 8 and b = 9, the operative binary operation is addition, hence a + b = 17 = b + a.
Associative property: If we can write (a* b) *c = a*(b * c) given a non-empty set A, we have the associative property of binary operations. Assume that N is a collection of natural integers, and that multiplication is a binary operation.a = 4, b = 5 ,c = 6. We can write (a* b) *c = 120 = a *(b *c).
Distributive property: Let A be a non-empty set with two binary operations * and o. If a*(b o c) = (a * b) o (a * c) or (b o c)*a = (b * a) o (c * a). The binary operations are distributive. Consider the symbols * and o to represent multiplication and subtraction, respectively. a = 2, b=5,c=4.
Then a*(b o c) = 2 (5 4) = 2 a*(b o c) = 2 a*(b o c) = 2 a*(b o c) =
a*(b o c) = a × (b – c) = 2 × (5 – 4) = 2.And (a * b) o (a * c) = (a × b) – (a × c) = (2 × 5) – (2 × 4) = 10 – 6 = 2.
Identity element: If A is a non-empty set and * is a binary operation on A, then If a * e = a = e * a, an element e is the identity element of an A. If the binary operation is addition (+), e equals 0, and if the binary operation is multiplication(*), e equals 1.
Inverse property: If a binary operation * on a set A that satisfies a * b = b * a = e is invertible for all a, b A. a-1 is invertible if a-1 = b for a * b = b * a = e. When * is multiplication, 1 is invertible.
The logical operation XOR, or eXclusive OR, compares the input values (bits) and generates the output value (bit). The exclusive OR logic is straightforward. The output is 0 if the input values are the same (or false). The outcome is 1 if the input values are different (or true).
To find XOR, we must first find both a and b’s binary representations. Let’s have a look at an example. Assume that a = 7 and b = 10. As a result, binary representations of a = 111 (as explained above) and b = 1010 are used.
Figure 1: logic symbol of XOR gate
Calculate the XOR of 2 numbers
1. First, we’ll convert both of the numbers to binary:
0101 0000 is the 8-bit binary equivalent of 80.
0110 0100 is the 8-bit binary representation of 100.
2. From the first to the last, we’ll use the rule to find the XOR of each pair of corresponding bits:
The output bit is 0 if both bits are the same, i.e. 1 (or 0).
The output is 1 if both bits are different.
3. The output bit will be 0 if the first bit pair is 0⊕0, We may determine the output bit for each pair in the same way.
In this article we learn that A binary operation can be thought of as a function with two elements from the same set S as input and an element from S as output. A pair (a,b) of elements in S can be represented as two elements a and b of S. and The binary number system is a variant of the decimal (10-base) number system that we are all familiar with. Binary numbers are significant because they simplify the design of computers and related technologies when compared to the decimal system.
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