Let S be a non-empty set and o be a binary operation on S defined by xoy - x-x-y - S- Determine whether o is commutative ans associative-

O is a binary operation on S x o y = x for x , y ∈ S For associative #160; #160;let x , y , z ∈ S ( x o y ) o z =( x o z )= x ...

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Standard IX

Mathematics

Associative Property

Let S be a ...

Question

Let $S$ be a non-empty set and $o$ be a binary operation on $S$ defined by $x o y = x : x, y \in S$ . Determine whether o is commutative ans associative.

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Solution

o is a binary operation on S
$x o y = x f o r x, y \in S$

For associative
let $x, y, z \in S (x o y) o z = (x o z) = x x o (y o z) = (x o y) = x \Rightarrow (x o y) o z = x o (y o z)$
So this binary operation is associative
For commutative
Let $x, y \in S x o y = x y o x = y x o y \neq y o x$
So this binary operation in not commutative

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Let S be a non-empty set and o be a binary operation on S defined by xoy=x:x,y∈S. Determine whether o is commutative ans associative.

o is a binary operation on Sxoy=xforx,y∈SFor associative let x,y,z∈S(xoy)oz=(xoz)=xxo(yoz)=(xoy)=x⇒(xoy)oz=xo(yoz)So this binary operation is associative For commutativeLetx,y∈Sxoy=xyox=yxoy≠yox So this binary operation in not commutative

Let $S$ be a non-empty set and $o$ be a binary operation on $S$ defined by $x o y = x : x, y \in S$ . Determine whether o is commutative ans associative.