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Identity element
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In mathematics, the identity element (or neutral element) of a set is a special element of that set. It is special because if it is combined with another element of that set, it does not change the other element.
With addition, the identity element is 0, because adding 0 to some number does not change the number. With multiplication, it is 1.
Further Examples
set  operation  identity 

real numbers  + (addition)  0 
real numbers  • (multiplication)  1 
real numbers  a^{b} (exponentiation)  1 (right identity only) 
mbyn matrices  + (addition)  zero matrix 
nbyn square matrices  • (multiplication)  identity matrix 
all functions from a set M to itself  ∘ (function composition)  identity map 
character strings, lists  concatenation  empty string, empty list 
extended real numbers  minimum/infimum  +∞ 
extended real numbers  maximum/supremum  ∞ 
subsets of a set M  ∩ (intersection)  M 
sets  ∪ (union)  {} (empty set) 
boolean logic  ∧ (logical and)  ⊤ (truth) 
boolean logic  ∨ (logical or)  ⊥ (falsity) 
only two elements {e, f}  * defined by e * e = f * e = e and f * f = e * f = f 
both e and f are left identities, but there is no right or twosided identity 

