7Arithmetics of Fuzzy Sets

Now, we want to introduce the arithmetic operations between fuzzy sets. Given any a comma b element-of double-struck upper R, we see that

chi Subscript left-brace a plus b right-brace Baseline left-parenthesis z right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column if z equals a plus b 2nd Row 1st Column 0 2nd Column if z not-equals a plus b period EndLayout equals sup Underscript StartSet left-parenthesis x comma y right-parenthesis colon z equals x plus y EndSet Endscripts min left-brace right-brace comma of chi chi left-brace right-brace a left-parenthesis right-parenthesis x comma of chi chi left-brace right-brace b left-parenthesis right-parenthesis y period

In general, let ring be any binary operations plus comma minus comma times comma slash between two real numbers a and b. Then, we have

(7.1)chi Subscript left-brace a ring b right-brace Baseline left-parenthesis z right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column if z equals a ring b 2nd Row 1st Column 0 2nd Column if z not-equals a ring b period EndLayout equals sup Underscript StartSet left-parenthesis x comma y right-parenthesis colon z equals x ring y EndSet Endscripts min left-brace right-brace comma of chi chi left-brace right-brace a left-parenthesis right-parenthesis x comma of chi chi left-brace right-brace b left-parenthesis right-parenthesis y comma

where b not-equals 0 when a slash b is considered.

7.1 Arithmetics of Fuzzy Sets in double-struck upper R

Let circled-dot denote any one of the four basic arithmetic operations between fuzzy sets and in with membership functions ...

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