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Question

# Consider the binary operations ∗:R×R→R and o:R×R→R defined as a∗b=|a−b| and a o b=a, ∀ a, b ∈ R. Show that ∗ is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c∈ R, a∗(b o c)=(a∗b)o(a∗c). [If it is so, we say that the operation ∗ distributes over the operation o]. Dose o distribute over ∗? Justify your answer

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Solution

## Check commutative for ∗∗ is commutative ifa∗b=b∗aa∗b=|a−b|;b∗a=|b−a|=|a−b|Since,a∗b=b∗a∀a,bϵR∴∗ is commutative.Check associative for ∗∗ is associative if(a∗b)∗c=(a∗b)∗c(a∗b)∗c=(|a−b|)∗c=||a−b|−c|a∗(b∗c)=a∗(|b−c|)=|a−|b−c||Since (a∗b)∗c≠a∗(b∗c)∗ is not associative.aob=aCheck commutative for 00 is commutative if, a0b=b0aa0b=aandb0a=bSince a0b≠b0a0 is not commutative.Check associative for 00 is associative if(a0b)0c=a0(b0c)(a0b)0c=a0c=aa0(b∗c)=a0b=aSince, (a0b)0c=a0(b0c)0 is not associative.a∗b=|a−b|anda0b=a0 distributes over ∗Ifa0(b∗c)=(a0b)∗(a0c),∀a,b,cϵR0 distributes over ∗a0(b∗c)=a0|b−c|(a0b)∗(a0c)=a∗a=|a−a|=|0|=0Sincea0(b∗c)≠(a0b)∗(a0c)0 does not distributes over ∗∗ distributes over 0Ifa∗(b0c)=(a∗b)0(a∗c),∀a,b,cϵR∗ distributes over 0.a∗(b0c)=a∗b=|a−b|(a∗b)0(a∗c)=|a−b|0|a−c|=|a−b|Sincea∗(b0c)=(a∗b)0(a∗c),∀a,b,cϵR∗ distributes over 0.

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