How to Show 'Not' Statements on a Venn DiagramDate: 05/03/2004 at 22:40:24 From: Rachel Subject: math tutoring How can I use a Venn diagram to display the opposite of a union or intersection of P & Q? For instance, how do I show (not P) or (not Q) versus not (P & Q)? I understand the positive aspect, I'm just not sure about the "NOT". Date: 05/04/2004 at 11:54:38 From: Doctor Ian Subject: Re: math tutoring Hi Rachel, Let's make a simple Venn diagram. (I'm using rectangles because circles are hard to make with a keyboard, but it's the same idea.) Note that I've included a universal set (U), which includes elements that might not be in either of the sets P or Q. +-----------U------------+ | | | +----Q----+ | | | | | | +------|----+ | | | | | | | | | | | | | | | | +---------+ | | | | | | | | | | +-----P-----+ | | | +------------------------+ Now, how can I represent "P and Q"? That's the intersection of the two sets, right? +-----------U------------+ | | | +----Q----+ | | | | | | +------|----+ | | | | |....| | | P and Q | | |....| | | | | +---------+ | | | | | | | | | | +-----P-----+ | | | +------------------------+ So what is "not (P and Q)"? It's everything else: +-----------U------------+ |........................| |.........+----Q----+....| |.........|.........|....| |..+------|----+....|....| |..|......| |....|....| not (P and Q) |..|......| |....|....| |..|......+---------+....| |..|...........|.........| |..|...........|.........| |..+-----P-----+.........| |........................| +------------------------+ Now, let's contrast this with '"not P) and (not Q)". In this case, we need to find everything that isn't in P, +-----------U------------+ |........................| |.........+----Q----+....| |.........|.........|....| |..+------|----+....|....| |..| | |....|....| not P |..| | |....|....| |..| +---------+....| |..| |.........| |..| |.........| |..+-----P-----+.........| |........................| +------------------------+ and then find everything that isn't in Q, +-----------U------------+ |........................| |.........+----Q----+....| |.........| |....| |..+------|----+ |....| |..|......| | |....| not Q |..|......| | |....| |..|......+---------+....| |..|...........|.........| |..|...........|.........| |..+-----P-----+.........| |........................| +------------------------+ and then intersect those two sets: +-----------U------------+ |........................| |.........+----Q----+....| |.........| |....| |..+------|----+ |....| |..| | | |....| (not P) and (not Q) |..| | | |....| |..| +---------+....| |..| |.........| |..| |.........| |..+-----P-----+.........| |........................| +------------------------+ Now, if we look at this last diagram, we can see that what we have, in fact, specified everything that is NOT in (P or Q): +-----------U------------+ |........................| |.........+----Q----+....| |.........| |....| |..+------|----+ |....| |..| | | |....| (not P) and (not Q) |..| | | |....| |..| +---------+....| |..| |.........| not (P or Q) |..| |.........| |..+-----P-----+.........| |........................| +------------------------+ That is to say, these expressions are equivalent: (not P) and (not Q) <=> not (P or Q) Try using the same technique to show that not (P and Q) <=> (not P) or (not Q) ? Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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