Identity Element

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Formal Definition

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Let $\langle S,*\rangle$ be a binary structure. An element $e$ of $S$ is an identity element for $*$ if $e*s=s*e=s$ for all $s\in S$.

Informal Definition

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An identity element is an element of a set with a binary operation on the set, that when combined with another element it leaves it unchanged.

Example(s)

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Let $*$ be defined on $\mathbb{Z}$ by letting $a*b=ab$. The identity element is 1, $a*1=1*a=a\hspace1cm b*1=1*b=b$.

Non-example(s)

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Let $*$ be defined on $\mathbb{Z^+}$ by letting $a*b=a+b$. There isn't an identity element because $0\notin\mathbb{Z^+}$.

Additional Comments

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