Some elementary set theory questions

[Townsend]

I am currently reviewing for an upcoming test over sets. What the instructor did was to give out the test he gave out for last semester for us to study from. I can answer most of these questions but there are a few that I am a little bit unsure of. Some of the questions are complete the definition and for these it is important that I get the definitions exactly right because he is rather picky about those. Anyways, it would be nice if you could look over the questions and my answers and tell me if I made any mistakes and possible help me out at various places.

1.Complete each definition. (the first part is what is given and then the second part is my answer that I put in parentheses.)

If A is a Set, Then the power set of A (i.e. P(A)) is (the set of all subsets of A.)

The Cartesian Product of sets A and B (i.e. A X B) is ({(a,b): a is an element of A, and b is an element of B})

If A is a set, then a partition of A is (a collection of nonempty, pairwise disjoint subsets of S, A1, A2, …, An, such that A1 U A2 U … U An=S)

The union of sets A and B is (the set of all elements in A or B.)

The intersection of sets A and B is (the set of all elements in A and B.)

The symmetric difference of sets A and B is ((AUB)-(A intersect B))

The compliment of A relative to B is (the set of elements in B that are not in A.)

Sets A and B are disjoint if (A intersect B is the empty set.)

Set A is a proper subset of set B if (every element in A is also and element of B and A is not equal to B.)

2. Let (U is the Universe) U={1,2,…,8} and let A={2,4,6}, B={1,3,5,7} and C={4,5,6}. Find the following.

A symmetric difference B = {1,2,3,4,5,6,7}

Compliment of B – A = {8}

(AUC)-B= {2,4,6}

(A-C) intersection (B-A) = {empty set}

3. Let U={1,2,3} and Let A={3} and B={1,3}. Find the following.

B X compliment of A = {(1,1), (1,2), (3,1), (3,2)}
Compliment of (compliment of A X B) = {(1,2), (2,2), (3,1), (3,2), (3,3)}

Compliment of A X B X compliment of B = {(1,1,2), (1,3,2), (2,1,2), 2,3,2)}

4. State the distributive law of union over intersection. State the associative law of intersection.

The distributive law of union over intersection is A U (B U C) = (A U B) intersection (AUC).

The associative law of intersection A intersect (B intersect C) = (A intersect B) intersect C.

5. True or False

The set {{1}, {2}, {3}, {4}} is a partition of the set {1,2,3,4}. (T)

Intersection is commutative. (T)

Relative compliment is commutative. (F)

The empty set is a proper subset of every set. (I think false but I am not sure. I say false because the empty set has the empty set as a subset of itself but the two sets are equal and hence one is not a proper subset of the other.)

Every set has at least one subset. (T)

{a} is an element of {a,b}. (F)

{a} is and element of {{a}, b} (T)

6. Fill in blank.

A set with four elements has 32 subsets. A set with three elements has 7 proper subsets.


That should work for now. Also would one of you who know it please give me a good definition of super set? Thanks

Best Regards

Jeremy

 

[Townsend]

Another question just came up...

true or false..

Set compliment is a binary operator

I cannot answer this because I am not sure what a binary operator is. I know binary means two and I know what is meant by operator. I think it means the operation is done on two sets or inputs. If my thinking is right then I think it would be ture.

Best regards

 

[cumin]

A binary operator assigns a member of a set to an ordered pair of elements from the set. The binary operator "+" assigns a real number to a pair of real numbers; 5.2 + 4.1 = 10.1. It is like a computer language method or function with two arguments.
Plus(5.2, 4.1) --> 10.1.

Negation is a unary operator, taking a single element and assigining a single element. For instance, Negation(4) = -4.

 

[Townsend]

That makes sense...

So for my example a set compliment operator takes a set and assigns it to another set which makes it a unary operation. For example if U={1,2,3,4,5} and A={1,2} then the set compliment of A would be the set {3,4,5}. And the set compliment operator assigned the set {3,4,5} to set A which is a unary operation.

I really appericate that explanation.

Best regards

Jeremy