Problem 66 Verify the assertion that two se... [FREE SOLUTION]

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Finite Mathematics for the Managerial, Life, and Social Sciences

Soo T. Tan

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 6: Problem 66

Verify the assertion that two sets $A$ and $B$ are equal if and only if (1) $A \subseteq B$ and (2) $B \subseteq A$.

Short Answer

Expert verified

To verify the assertion that two sets A and B are equal if and only if (1) $A \subseteq B$ and (2) $B \subseteq A$, we prove two implications: 1. If A and B are equal, then $A \subseteq B$ and $B \subseteq A$. This is true because if $A = B$, every element of A is an element of B and vice versa. 2. If $A \subseteq B$ and $B \subseteq A$, then A and B are equal. This is true since every element in A is an element in B and every element in B is an element in A. Hence, we have proven that sets A and B are equal if and only if (1) $A \subseteq B$ and (2) $B \subseteq A$.

Step by step solution

Prove that if A and B are equal, then $A \subseteq B$ and $B \subseteq A$

Assume that $A = B$. By definition of set equality, every element of A is also an element of B, and every element of B is also an element of A. This confirms that $A \subseteq B$ and $B \subseteq A$.

Prove that if $A \subseteq B$ and $B \subseteq A$, then A and B are equal

Now, let's assume that $A \subseteq B$ and $B \subseteq A$. We need to show that A and B have the same elements, and therefore they are equal sets. First, let $x$ be an arbitrary element in A. Since $A \subseteq B$, it must be that $x$ is also an element of B. Hence, every element in A is also an element in B. Next, let $y$ be an arbitrary element in B. Since $B \subseteq A$, it must be that $y$ is also an element of A. Hence, every element in B is also an element in A. Since every element in A is an element in B and every element in B is an element in A, we can conclude that A and B are equal sets. This proves the second implication.

Conclusion

By proving the two implications, we have verified the assertion that two sets A and B are equal if and only if (1) $A \subseteq B$ and (2) $B \subseteq A$.

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91影视

Verify the assertion that two sets \(A\) and \(B\) are equal if and only if (1) \(A \subseteq B\) and (2) \(B \subseteq A\).

Short Answer

Step by step solution

Prove that if A and B are equal, then \(A \subseteq B\) and \(B \subseteq A\)

Prove that if \(A \subseteq B\) and \(B \subseteq A\), then A and B are equal

Conclusion

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