*
The binary operation'*' defined on the set of integer as a*b= |a+b|+5 is
Commutative
Non commutative
Associative but Non commutative
Associative
Answers
in mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it.
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In the given options correct option is not mentioned. Please refer to the detailed solution and correct option.
Answer:
The given condition a*b=|a+b|+5 satisfies both Associative law and Commutative law.
Step-by-step explanation:
Binary operation:
Binary operation means when a mathematical operation is performed on any two elements of a set, then the output also belongs to the same set.
The binary operation can be of the form *, +, -
Associative law:
Let a, b, c ∈ Z(set of integers) with binary operation '*' then Associative law is defined as a*(b*c)=(a*b)*c
Commutative law:
Let a,b ∈ Z(set of integers) with binary operation '*' then Commutative law is defined as a*b = b*a
- Given a*b=|a+b|+5 and * is defined on the set of integers.
Now we have to check whether the above-given condition
satisfies the laws or not.
- For a, b, c ∈ Z, a*b=|a+b|+5
- Now check a*(b*c)=(a*b)*c
Consider
L.H.S = a*(b*c) = a*(|b+c|+5)
= a*(b+c+5)
= |a+b+c+5|+5
= a+b+c+10
R.H.S = (a*b)*c = (|a+b|+5)*c
= (a+b+5)*c
= |a+b+5+c|+5
= a+b+c+5+5
= a+b+c+10
Since, L.H.S = R.H.S
So, the given condition satisfies Associative law.
- For a, b ∈ Z, a*b=|a+b|+5
- Check a*b = b*a
Consider
L.H.S= a*b = |a+b|+5
= a+b+5
R.H.S = b*a = |b+a|+5
= b+a+5 ( Since a, b ∈ Z, a+b = b+a)
Since L.H.S = R.H.S
So, the given condition satisfies Commutative law.
Hence the given condition a*b=|a+b|+5 satisfies both Associative law and Commutative law.
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