Set_Arithmetic

Set_Arithmetic - ITI 111 1/28/2011 Lecture 4 notes Set...

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Unformatted text preview: ITI 111 1/28/2011 Lecture 4 notes Set Arithmetic Review of concepts 5 +7 = 7 +5 3 x8 = 8 x3 So we say multiplication and addition are “commutative ” (7 + 5) + 3 = (3 + 5) + 7 (7 x 2) x 3 = 7 x (2 x 3) So we say that multiplication i s “associative ” 7 x (3 + 2) = (7 x 3) + (7 x 2) So we say multiplication is “distributive ” with respect to addition. Note that addition is not distributive with respect to multiplication. Set Arithmetic What about sets? So both union and intersection are commutative ( ) ( ) ( ) ( ) So both union and intersection are associative ( ) ( ) ( ) It turns out that intersection is distributive with respect to un ion. To prove this: or And e ither way… It is also true that union is distributive with respect to union. We can demonstrate this using the same procedure. However: ( ) ( ) And therefore union and intersection are not associative. To see this, assume that C is the empty set and A i s any non-empty set. In this case the left hand side would reduce to the empty set, whi le the right hand side would reduce to A. ...
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