Hello! I would appreciate some help on a few questions. Many thanks!! 1. Is an \

Question

Hello! I would appreciate some help on a few questions. Many thanks!!

1. Is an "element of a set" a special case of a "subset of a set"?

2. List all subsets of {0,1,3}. How many do you get?

3. Prove that l A U B I + I A intersect B I  = I A I + I B I

4. a.) What is the symmetric difference of the set Z+ of nonnegative integers and the set E of even integers (E = {...,-4,-2,0,2,4,...} contains both negative and positive even integers).

b.) Form the symmetric difference of A and B to get a set C. Form the symmetric difference of A and C. What did you get? Give a proof of the answer.

Explanation / Answer

1. yes , it is the special case. a element of a set if taken as a set , then it becomes the subset of the set.

2. {0}, {1}, {3}, {0,1}, {0,3}, {1,3}, empty set 'phi' , and {0,1,3}, hence we get 8 subsets.

3. if we add number of elements in A and number of elements in B, then we count the number of elemetns in A intersection B twice. 1st while counting in A we count the common elements, then again in B we count the common elements, hence we need to subtract the number of common elements once from the sum |A|+|B|

therefore. |A|+|B|-| A intesection B|= |A U B|

4.symmetric difference means the elemnts which are in A or B but in not both. ie. not in A intersection B

here intersection of E and Z+ are the all nonnegative even integers. thus symmetric difference will have all the negative even integers and all the postive odd integers.

5. A-B=C i.e C= A intesection B' (1), Hence

A-C= A intesection C'=A intesection ( A intesection B')' from above equation (1),

=A intersection ( A' U B) from de morgans law, and

= (A intersection A' ) U (A intersection B)

= A intersection B

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