Another special set of sets is the set that contains all the possible combinatio

Question

Another special set of sets is the set that contains all the possible combinations of the elements in the set. In this case there may be duplicates. For the set A the set of all possible subsets is: {/0, {1}, {2}, {1,2}, {1,3}, {2,3}, {1,2,3}}
a) What is the first element in the set of all subsets? b) How many elements are in the set of all subsets? c) Are any of the elements of set A duplicated within any of the sets of all the possible subsets?

2. Complete the following for the set B = {2, 4, 6, 8}:
a) P(B) = {
} b) How many elements in P(B)?

3. Let A = {2, 3, 4} and B = {3, 5}. How many elements are in each of the following sets?
a) P(A) b) P(B) c) P(A?B)



PICTURE IS FOR 4-6
4. a) To build the power set we started with what power set? b) What is {1} U /0? c) After adding 1, What is the power set? d) What is {2} U /0? What is {2} U {1}? e) After adding 2, What is the power set? f) After adding 3, What is the power set?
5. a) What would be the power set after adding 4? b) What is n(/0), the cardinal it’s (number of elements) of the empty set? What n(P(/0)) the cardinal it’s of the power set of the empty set? What is n(P(/0)) expressed as a power of 2( 2 to some power)? c) What is n({1})? What is n(P({1})) expressed as a power of 2? d) What is n({1,2})? What n({P({1,2})) expressed as a power of 2? e) n({1,2,3})? n(P({1,2,3})) expressed as a power of 2? f) n(P({1,2,3,4})) expressed as a power of 2? g) What would be the gene4al equation for the cardinal it’s of a power set for set S with k elements? n(P(S)) =
6. Let’s eat, we have four flavors of ice cream: banana, chocolate, lemon, and strawberry. How many different ways can we have them? (Assuming at most one scoop per flavor)
Let’s use letters for the flavors: (B, C, L, S). First, how many possible combinations are there?
List all the possible combinations. [Remember, you might have a dieter who takes no ice cream]



Activity 15-Power Sets and Partitions Model III. (15 min) Building a Power Set: To build a power set for the superset: D (1, 2, 3, 4) we can use the following algorithm. . Start with an empty set, , and its power-set, (o) .For every element inside the superset: 1. Create a copy of every set in the current power set 2. Add (union) the element to each one of the current power set. 3. Add the copies to the current power set. After the first iteration we added 1 to 0 giving [1) After the second iteration (0, (1), 12), (1,2) we added 2 to Ogiving (2) and 2 to (1) giving [1,2) After the third iteration

Explanation / Answer

1)

Answer)

A={1,2,3}

The set of all possible subsets : {/0,{1},{2},{1,2},{1,3},{2,3},{1,2,3}}

The first element in the set of all subsets is the empty set or the null set which is denoted by /0.

There are 7 elements in the set of all subsets are: /0,{1},{2},{1,2},{1,3},{2,3},{1,2,3}

Yes, 2 elements of Set A such as 1,2 are duplicated with the elements of the sets of all the possible subsets. This is as: A = {1,2,3} and sets of all the possible subsets is {/0, {1}, {2}, {1,2}, {1,3}, {2,3}, {1,2,3}}

2)

Answer)

set B = {2, 4, 6, 8}:

P(B)={{}, {2}, {4}, {2, 4}, {6}, {2, 6}, {4, 6}, {2, 4, 6}, {8}, {2, 8}, {4, 8}, {2, 4, 8}, {6, 8}, {2, 6, 8}, {4, 6, 8}, {2, 4, 6, 8}}

P(B) has 16 elements.

3)

A = {2, 3, 4} and B = {3, 5}

P(A)={{}, {2}, {3}, {2, 3}, {4}, {2, 4}, {3, 4}, {2, 3, 4}}

P(A) has 8 elements.

P(B)={{}, {3}, {5}, {3, 5}}

P(B) has 4 elements.

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