Properties of Binary Operations

IMPORTANT

Properties of Binary Operations: Overview

This topic covers concepts, such as, Laws of Binary Operations, Closure Property of Binary Operations, Existence of Inverse for a Binary Operation & Existence of Non-zero Divisors for a Binary Operation etc.

Important Questions on Properties of Binary Operations

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Explain the inverse of binary operation.

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Define a binary operation * on the set 0,1,2,3,4,5 as
a*b=a+bif a+b<6a+b-6if a+b6
Show that zero is the identity for this operation and each element a0 of the set is invertible with 6-a being the inverse of a.

HARD
IMPORTANT

Given a non-empty set X, consider the binary operation * : PX×PXPX given by A*B=AB  A,B in PX, where PX is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in PX with respect to the operation *.

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Let A=N×N and * be the binary operation on A defined by a,b*c,d=a+c,b+d. Show that * is commutative and associative. Find the identity element for * on A, if any.

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Let * be a binary operation on the set Q of rational numbers as a*b=a+ab. Find whether the binary operation is commutative or associative or both.

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Let * be a binary operation on the Q of rational numbers,

a*b=a2+b2

 Then find out the binary operation is

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Show that division is not a binary operation on real numbersR. Give one example.

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Show that subtraction and division are not binary operations on N.

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a*b is a binary operation, a=b. Verify reflexive property for given binary operation.

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a*b is a binary operation, a<b. Verify reflexive property for given binary operation.

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a*b is a binary operation, ab. Verify reflexive property for given binary operation.

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a*b is a binary operation, ab. Verify reflexive property for given binary operation.

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a*b is a binary operation, a>b. Verify reflexive property for given binary operation.

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Suppose * be the multiplication operation and # be the addition operation defined on Z. Let a=10,b=11 and c=12, then find a*(b#c).

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Suppose * be the multiplication operation and # be the addition operation defined on Z. Let a=5,b=6 and c=8, then find a*(b#c).

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Suppose * be the multiplication operation and # be the addition operation defined on Z. Let a=3,b=4 and c=7, then find a*(b#c).

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Suppose * be the multiplication operation and # be the subtraction operation defined on Z. Let a=5,b=6 and c=8, then find a*(b#c).

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Suppose * be the multiplication operation and # be the subtraction operation defined on Z. Let a=3,b=4 and c=7, then find a*(b#c).

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Let a*b=L.C.M (a,b) be the binary operation defined on the set of natural numbers, where the identity element is e=1. Find the number of invertible elements.

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Let * be a binary operation on Q, defined by a*b=ab2, a, bQ. Thus * is associative.