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A binary operation Ηon a set of integers is defined as x⊙y=x2+y2+2xy. Which one of the following statements is true aboutΗ?
- a)Commutative but not associative
- b)Both commutative and associative
- c)Associative but not commutative
- d)Neither commutative nor associative
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A binary operation Ηon a set of integers is defined as x⊙y=x2+y2+2xy....
1.Check for commutative
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(x⊙y)=(y⊙x)
x2 + y2 + 2xy = y2 + x2 + 2yx
Since LHS = RHS
So ⊙ is commutative.
2. Check for associative:
(x⊙y)⊙z=x⊙(y⊙z)
(x2 + y2 + 2xy) Η z = x Η ( y2 + z2 + 2yz)
x4 + y. + 4x2y2 + 2x2y2 + 4xy3 + 4x3y + z2 + 2x2z + 2y2z + 4xyz ≠ x2 + y4 + z4 + 4y2z2 + 2y2z2 + 4y3z + 4yz3 + 2xy2 + 4xyz
So A is not associative.
Most Upvoted Answer
A binary operation Ηon a set of integers is defined as x⊙y=x2+y2+2xy....
To determine whether the binary operation ⊙ is commutative or associative, we need to examine its properties.
Commutative Property:
A binary operation ⊙ is commutative if for any two elements x and y in the set, x⊙y = y⊙x. In other words, the order of the operands does not affect the result.
Associative Property:
A binary operation ⊙ is associative if for any three elements x, y, and z in the set, (x⊙y)⊙z = x⊙(y⊙z). In other words, the grouping of the operands does not affect the result.
Let's analyze the given operation ⊙ to determine whether it satisfies these properties.
Commutative Property:
To check if ⊙ is commutative, we need to verify if x⊙y = y⊙x holds true for any x and y in the set.
Let's take arbitrary values for x and y and evaluate both sides of the equation:
x⊙y = x^2 * y^2 / (2xy)
y⊙x = y^2 * x^2 / (2yx)
If we simplify both expressions, we get:
x⊙y = y⊙x = xy
Since xy is not always equal to x⊙y, the operation ⊙ is not commutative.
Associative Property:
To check if ⊙ is associative, we need to verify if (x⊙y)⊙z = x⊙(y⊙z) holds true for any x, y, and z in the set.
Let's take arbitrary values for x, y, and z and evaluate both sides of the equation:
(x⊙y)⊙z = ((x^2 * y^2) / (2xy))⊙z = (((x^2 * y^2) / (2xy))^2 * z^2) / (2((x^2 * y^2) / (2xy))z)
x⊙(y⊙z) = x⊙((y^2 * z^2) / (2yz)) = (x^2 * ((y^2 * z^2) / (2yz))^2) / (2xy((y^2 * z^2) / (2yz)))
If we simplify both expressions, we get:
(x⊙y)⊙z = x⊙(y⊙z) = xyz
Since xyz is not always equal to (x⊙y)⊙z, the operation ⊙ is not associative.
Therefore, the correct answer is option (A) - Commutative but not associative.
Commutative Property:
A binary operation ⊙ is commutative if for any two elements x and y in the set, x⊙y = y⊙x. In other words, the order of the operands does not affect the result.
Associative Property:
A binary operation ⊙ is associative if for any three elements x, y, and z in the set, (x⊙y)⊙z = x⊙(y⊙z). In other words, the grouping of the operands does not affect the result.
Let's analyze the given operation ⊙ to determine whether it satisfies these properties.
Commutative Property:
To check if ⊙ is commutative, we need to verify if x⊙y = y⊙x holds true for any x and y in the set.
Let's take arbitrary values for x and y and evaluate both sides of the equation:
x⊙y = x^2 * y^2 / (2xy)
y⊙x = y^2 * x^2 / (2yx)
If we simplify both expressions, we get:
x⊙y = y⊙x = xy
Since xy is not always equal to x⊙y, the operation ⊙ is not commutative.
Associative Property:
To check if ⊙ is associative, we need to verify if (x⊙y)⊙z = x⊙(y⊙z) holds true for any x, y, and z in the set.
Let's take arbitrary values for x, y, and z and evaluate both sides of the equation:
(x⊙y)⊙z = ((x^2 * y^2) / (2xy))⊙z = (((x^2 * y^2) / (2xy))^2 * z^2) / (2((x^2 * y^2) / (2xy))z)
x⊙(y⊙z) = x⊙((y^2 * z^2) / (2yz)) = (x^2 * ((y^2 * z^2) / (2yz))^2) / (2xy((y^2 * z^2) / (2yz)))
If we simplify both expressions, we get:
(x⊙y)⊙z = x⊙(y⊙z) = xyz
Since xyz is not always equal to (x⊙y)⊙z, the operation ⊙ is not associative.
Therefore, the correct answer is option (A) - Commutative but not associative.
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A binary operation Ηon a set of integers is defined as x⊙y=x2+y2+2xy. Which one of the following statements is true aboutΗ?a)Commutative but not associativeb)Both commutative and associativec)Associative but not commutatived)Neither commutative nor associativeCorrect answer is option 'A'. Can you explain this answer?
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A binary operation Ηon a set of integers is defined as x⊙y=x2+y2+2xy. Which one of the following statements is true aboutΗ?a)Commutative but not associativeb)Both commutative and associativec)Associative but not commutatived)Neither commutative nor associativeCorrect answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A binary operation Ηon a set of integers is defined as x⊙y=x2+y2+2xy. Which one of the following statements is true aboutΗ?a)Commutative but not associativeb)Both commutative and associativec)Associative but not commutatived)Neither commutative nor associativeCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A binary operation Ηon a set of integers is defined as x⊙y=x2+y2+2xy. Which one of the following statements is true aboutΗ?a)Commutative but not associativeb)Both commutative and associativec)Associative but not commutatived)Neither commutative nor associativeCorrect answer is option 'A'. Can you explain this answer?.
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