ECE462Lect8 - Func%on Classica%on Symmetric Func%ons A...

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Func%on’Classifca%on’
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Symmetric’Func%ons’ A’func%on’f’is’ totally symmetric ’if’and’only’if’ it’is’unchanged’by’any’permuta%on’of’its’ variables’ f’=’ xy’+’ xz’+’ yz’ f’=’ xy±’+’ x±y’ A’func%on’f’is’ totally symmetric ’if’and’only’if’ it’can’be’speci²ed’by’sta%ng’a’list’of’integers’ A’=’{a 1 ,’a 2 ,…,a m },’0’≤’j’≤’n’so’that’f’=’1’if’and’ only’if’exactly’j’of’the’variables’are’1’ f
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Mixed’Symmetric’Func%ons’ A’func%on’f’is’ mixed symmetric ’if’and’only’if’ it’is’not’totally’symmetric,’but’it’can’be’ changed’into’a’totally’symmetric’func%on’by’ replacing’some’of’its’variables’by’their’ complements’ f’=’ xy±z±’ Replace’y±’by’y’and’z±’by’z’
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Symmetric’Func%ons’ A’func%on’f’is’ symmetric ’if’it’is’either’totally’ or’mixed’symmetric’
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Posi%ve’and’Nega%ve’Func%ons’ A’func%on’f(x 1 ,’x 2 ,…,x n )’is’ posi0ve ’in’x i ’if’and’ only’if’it’is’possible’to’write’a’sum-of-products’ expression’for’f’in’which’x i
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ECE462Lect8 - Func%on Classica%on Symmetric Func%ons A...

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