Consider the following statement. Assume that all sets are subsets of a universal set U. For all sets A and B, if A CB then 8°CA. Use an element argument to construct a proof for the statement by putting selected sentences from the following scrambled list in the correct order. By definition of complement, xE B. Suppose A and B are any sets such that A C B, and suppose x E B". Suppose A and B are any sets such that A C B, and suppose x E B. Therefore, by definition of complement x € A°, and thus, by definition of subset, B c A°. If x were in A, then x would have to be in B by definition of subset. But xe B, and so x E A. Hence, x E A, because A NB = 0. Proof: 1. ---Select--- 2. ---Select- 3. -Select--- 4. -Select--- Need Help? Read It Submit Answer
Consider the following statement. Assume that all sets are subsets of a universal set U. For all sets A and B, if A CB then 8°CA. Use an element argument to construct a proof for the statement by putting selected sentences from the following scrambled list in the correct order. By definition of complement, xE B. Suppose A and B are any sets such that A C B, and suppose x E B". Suppose A and B are any sets such that A C B, and suppose x E B. Therefore, by definition of complement x € A°, and thus, by definition of subset, B c A°. If x were in A, then x would have to be in B by definition of subset. But xe B, and so x E A. Hence, x E A, because A NB = 0. Proof: 1. ---Select--- 2. ---Select- 3. -Select--- 4. -Select--- Need Help? Read It Submit Answer
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider the following statement. Assume that all sets are subsets of a universal set \( U \).
For all sets \( A \) and \( B \), if \( A \subseteq B \) then \( B^C \subseteq A^C \).
Use an element argument to construct a proof for the statement by putting selected sentences from the following scrambled list in the correct order.
1. By definition of complement, \( x \notin B \).
2. Suppose \( A \) and \( B \) are any sets such that \( A \subseteq B \), and suppose \( x \in B^C \).
3. Suppose \( A \) and \( B \) are any sets such that \( A \subseteq B \), and suppose \( x \in B \).
4. Therefore, by definition of complement \( x \in A^C \), and thus, by definition of subset, \( B^C \subseteq A^C \).
5. If \( x \) were in \( A \), then \( x \) would have to be in \( B \) by definition of subset. But \( x \notin B \), and so \( x \notin A \).
6. Hence, \( x \in A \), because \( A \cap B = \emptyset \).
**Proof:**
1. Suppose \( A \) and \( B \) are any sets such that \( A \subseteq B \), and suppose \( x \in B^C \).
2. By definition of complement, \( x \notin B \).
3. If \( x \) were in \( A \), then \( x \) would have to be in \( B \) by definition of subset. But \( x \notin B \), and so \( x \notin A \).
4. Therefore, by definition of complement \( x \in A^C \), and thus, by definition of subset, \( B^C \subseteq A^C \).
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Transcribed Image Text:Consider the following statement. Assume that all sets are subsets of a universal set \( U \).
For all sets \( A \) and \( B \), if \( A \subseteq B \) then \( B^C \subseteq A^C \).
Use an element argument to construct a proof for the statement by putting selected sentences from the following scrambled list in the correct order.
1. By definition of complement, \( x \notin B \).
2. Suppose \( A \) and \( B \) are any sets such that \( A \subseteq B \), and suppose \( x \in B^C \).
3. Suppose \( A \) and \( B \) are any sets such that \( A \subseteq B \), and suppose \( x \in B \).
4. Therefore, by definition of complement \( x \in A^C \), and thus, by definition of subset, \( B^C \subseteq A^C \).
5. If \( x \) were in \( A \), then \( x \) would have to be in \( B \) by definition of subset. But \( x \notin B \), and so \( x \notin A \).
6. Hence, \( x \in A \), because \( A \cap B = \emptyset \).
**Proof:**
1. Suppose \( A \) and \( B \) are any sets such that \( A \subseteq B \), and suppose \( x \in B^C \).
2. By definition of complement, \( x \notin B \).
3. If \( x \) were in \( A \), then \( x \) would have to be in \( B \) by definition of subset. But \( x \notin B \), and so \( x \notin A \).
4. Therefore, by definition of complement \( x \in A^C \), and thus, by definition of subset, \( B^C \subseteq A^C \).
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