course Mth 151 ‹¾_’]¤Ã—Ü©§‡ŽÅðxžäß³rêû¹Çassignment #002
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13:36:23 2.2.24 U={a,b,...,g}, A={a,e}, B={a,b,e,f,g}, C={b,f,g}, D={d,e} Is C ps U? Is the statement true or false and why?
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RESPONSE --> The statement is true, because every element in subset C is in U, and there are some in U that are not in C. So the statement is TRUE. confidence assessment: 3
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13:36:27 ** It's true because all elements of C are in the universal set, and because there are elements of U that aren't in C. You have to have both conditions, since a proper subset cannot be identical to other set. **
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RESPONSE --> OK self critique assessment: 3
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13:39:26 Query 2.2.30 phi s D
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RESPONSE --> That is true. phi, or null, is always a subset of any set, or subset for that matter. So this statement is true. I arrived at this answer by looking at an earlier problem, which included phi as a proper subset, a subset, and the compliment of itself, so it can be many things. confidence assessment: 2
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13:40:13 ** Note that You should be responding to problem 2.2.30 from the homework you worked out on paper. The shorthand notation is for my reference and won't always make sense to you. For clarification, though, the symbol for the empty set is the Greek letter phi. One set is a subset of another if every element of that set is in the other. To show that a set isn't a subset of another you have to show something in that set that isn't in the other. There's nothing in the empty set so you can never do this--you can never show that it has something the other set doesn't. So you can never say that the empty set isn't a subset of another set. Thus the empty set is a subset of any given set, and in particular it's a subset of D. ALTERNATIVE ANSWER: As the text tells you, the empty set is a subset of every set. ANOTHER ALTERNATIVE Every element of the empty set is in D because there is no element in the empty set available to lie outside of D. ONE MORE ALTERNATIVE: The empty set is a subset of every set. Any element in an empty set is in any set, since there's nothing in the empty set to contradict that statement. **
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RESPONSE --> I acheived the alternative answer, but I will keep the notes for future reference. self critique assessment: 2
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13:41:43 2.2.33 D not s B Is the statement true or false and why?
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RESPONSE --> It's true, D cannot possibly be a subset of B, they don't even share the same elements. Element d appears in D, but not in B, how can one make a subset with elements it doesn't have? So the answer is TRUE. confidence assessment: 3
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13:41:47 ** D is a subset of B if every element of D is an element of B-i.e., if D doesn't contain anything that B doesn't also contain. The statement says that D is not a subset of B. This will be so if D contains at least one element that B doesn't. **
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RESPONSE --> self critique assessment: 3
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13:42:59 2.2.36 there are exactly 31 subsets of B
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RESPONSE --> False, there are many more possibilties of B. They can have sets that share elements, but aren't neccesarily in the same order. So the statement is FALSE. confidence assessment: 3
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13:45:05 ** If a set has n elements then is has 2^n subsets, all but one of which are proper subsets. B has 5 elements so it has 2^5 = 32 subsets. So the statement is false. There are exactly 31 proper subsets of B, but there are 32 subsets of B. **
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RESPONSE --> I got the answer right, but the reason I arrived at it was incorrect, which I am disappointed with. I wrote out every single possibility of a subset I could come up with, and simply counted them out. Remembering that finding the subsets of a set uses 2^n And that simply finding the amount of proper subsets, which is what you get if you don't include phi as a subset is 2^n-1. self critique assessment: 2
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13:48:09 Query 2.2.40 there are exactly 127 proper subsets of U Is the statement true or false and why?
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RESPONSE --> The statement is true. If we use 2^7 to find the amount of subsets, it gives us: 128. So finding proper subsets, we use: 2^7-1 128-1= 127. So the statement is true, there ARE 127 proper subsets. I am confident of this answer, as my self-critque and introspection upon the way I answered the previous question led me to this solution. confidence assessment: 3
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13:48:17 ** The set is not a proper subset of itself, and the set itself is contained in the 2^n = 2^7 = 128 subsets of this 7-element set. This leaves 128-1 = 127 proper subsets. **
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RESPONSE --> OK self critique assessment: 3
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13:50:54 Query 2.2.48 U={1,2,...,10}, complement of {2,5,7,9,10} What is the complement of the given set?
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RESPONSE --> The compliment of the given set is: (1,3,4,6,8) because those do not appear in the given set. at first, I was very confused, because this is question 50, not question 48 in the book. confidence assessment: 3
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13:50:59 ** the complement is {1,3,4,6,8}, the set of all elements in U that aren't in the given set. **
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RESPONSE --> self critique assessment: 3
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13:53:50 query 2.2.63 in how many ways can 3 of the five people A, B, C, D, E gather in a suite?
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RESPONSE --> 10 ways. {A,B,C}. {A.B,D}. {A,B,E}, {A,C,D}, {A,C,E}, {A,D,E}, {B,C,D}, {B,C,E}, {B,D,E}, {C,D,E}. Those are all the possibilties you receive when you use 3 numbers of the five elements in the set. confidence assessment: 3
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13:54:00 ** The answer here would consist of a list of all 3-element subsets: {a,b,c}, {a,b,d}, {a,b,e}, {a,c,d} etc. There are ten such subsets. Using a,b,c,d,e to stand for the names, we can list them in alphabetical order: {a,b,c), {a,b,d}, {a,b,e}, {a,c,d}, {a,c,e}, {a,d,e|, {b,c,d}, {b,c,e}, {b,d,e}, {c, d, e}**
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RESPONSE --> OK self critique assessment: 3
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