course Mth
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19:55:20 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 --> True C's elements are a part of or contained in set U, but it is not identical/equal so it is a proper subset.
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19:55:38 ** 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
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19:57:21 Query 2.2.30 phi s D
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RESPONSE --> True All sets have a subset of themselves and of the empty set. So the empty set is a subset of set D.
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20:00:03 ** 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 understood very little of the explanation. However I know that I was correct in that the empty set is a subset of every set.
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20:01:54 2.2.33 D not s B Is the statement true or false and why?
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RESPONSE --> True The elements in set D are not all in set B. D contains d and e. Only e is in set B. Since not all elements of D are in B, D is not a subset of B.
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20:02:27 ** 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 --> ok
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20:04:30 2.2.36 there are exactly 31 subsets of B
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RESPONSE --> False There are 32 subsets of B. Using the Powers of 2 and knowing that there are 5 elements in set B this tells us that there are 32, not 31 subsets in set B. There are 31 proper subsets of B.
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20:04:59 ** 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 --> ok
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20:06:55 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 --> True There are 7 elements in U, so we say 2^7 and come up with 128. To find the proper subsets we take one away (2^7 - 1). The answer is 127, so the statement is true.
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20:07:09 ** 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
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20:09:33 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 --> {1, 3, 4, 6, 8} The complement of a set is its opposite or what is not included. U is 1-10 and the complement of using numbers 2, 5, 7, 9, 10 are the other numbers not stated.
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20:09:41 ** 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 --> ok
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20:15:25 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 --> {ABC}, {ABD}, {ABE}, {ACD}, {ACE}, {ADE}, {BCD}, {BCE}, {BDE}, {CDE}
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20:15:47 ** 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
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