logic 101 (class 3, set)

Somehow, "half-knowledge (mis-understood) is more dangerous then foolish". it is because, people like to guess the non-told part, or be confused for the non-told. lets go deeper into logic.

before, lets go back to a fundamental thing : set.

    set is a group of objects

for example, a family is a set, it contains family members. the stuffs on a desk form a set. the people in a country form a set. nevertheless, there is an empty set. say, the tree in Antarctica. Thus, by naming the common property of some objects or a rule on selection, we have a set. in reverse, all elements in a set share some common properties or rules.

among a set, there may be some "sub-set", say, the kids in a family, the women in a country. this concept should be easily be understood. 

To get the feeling of set, we usually draw a picture. the area inside a circle represent a set, and another circle inside the circle represent a sub-set.

Now, we have to introduce 2 simple concepts: AND and OR

     AND : the common objects in 2 or more sets.

     OR : any thing in 2 or more sets

in a picture, AND is the common area of 2 circles. OR is the total area of 2 circles. We should be very careful that, in logic, the OR has only 1 meaning. but in human language, which also follow some logic, "or" can be mean "either A or B, but not both", or "either A or B or both." 

the application of it is very powerful. say, we have a set of  "what to do in raining", thus, we have many elements inside, say, "bring umbrella" , "stay indoor", etc. If we draw a circle A of  "what to do in raining", the circle B of "bring umbrella" is another circle that cross circle A. and "stay indoor" also another circle crossing A but never touch B. 

what does it means?? it mean that, there are some are of B, which is not lay in A. thus, when someone bring an umbrella, it does not mean it is raining! we just illustrated the logic why "If A then B" and "if B then A" are not equal by graph!! 

and we can see, the circle B and C don't touch each other, which mean, when we stay home, we never bring umbrella and vice versa. then we have a logic, 

    if B then not C   equal   if C then not B

ok, Now, we introduce "a universal set", which contain everything we are concerning. (it is not the set of everything, which does not exist.) we draw a rectangle which contain every circles.;thus, the area OUTSIDE  a circle mean "not this circle". in our example, "not C" is outside the circle C.

thus, we can see, B is a subset of not C. and we have a very important concept.

   if A then B  equal    set A is a sub-set of B

OK, lets apply on our case. we immediately see some problem that, circle A is not a subset of circle B! how come?  this mean, our logic has problem. we missing an assumption that, we have to add a circle D, "out-door". i will let reader think about where the "shape" (not circle may be) should be placed.