Explanation Let set of natural numbers N and set N ∪ { 0 } . It clear that N ∅ N ∪ { 0 } . The two sets have same cardinality if and only if there is one-to-one correspondence. That is if there exists one-to-one and onto function between them. A function is defined as f : N → N ∪ { 0 } by f ( n ) = n − 1 . For one-to-one we have to show that if f ( n 1 ) = f ( n 2 ) ⇒ n 1 = n 2

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Chapter 3.3, Problem 6E

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Whether the statement “If A⊂≠B, then A and B do not have the same cardinality.” is true or false.

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Chapter 3.3, Problem 5E

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5. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.5.AE.003. y y= ex² 0 Video Example x EXAMPLE 3 (a) Use the Midpoint Rule with n = 10 to approximate the integral कर L'ex² dx. (b) Give an upper bound for the error involved in this approximation. SOLUTION 8+2 1 L'ex² d (a) Since a = 0, b = 1, and n = 10, the Midpoint Rule gives the following. (Round your answer to six decimal places.) dx Ax[f(0.05) + f(0.15) + ... + f(0.85) + f(0.95)] 0.1 [0.0025 +0.0225 + + e0.0625 + 0.1225 e0.3025 + e0.4225 + e0.2025 + + e0.5625 €0.7225 +0.9025] The figure illustrates this approximation. (b) Since f(x) = ex², we have f'(x) = 0 ≤ f'(x) = < 6e. ASK YOUR TEACHER and f'(x) = Also, since 0 ≤ x ≤ 1 we have x² ≤ and so Taking K = 6e, a = 0, b = 1, and n = 10 in the error estimate, we see that an upper bound for the error is as follows. (Round your final answer to five decimal places.) 6e(1)3 e 24( = ≈

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Chapter 3 Solutions

Discrete Mathematics with Graph Theory

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Whether the statement “If A ⊂ ≠ B , then A and B do not have the same cardinality.” is true or false.

Solution Summary: The author analyzes whether the statement "If Aundersetne subset >B, then A and B do not have the same cardinality" is true or false.

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Whether the statement “If A⊂≠B, then A and B do not have the same cardinality.” is true or false.

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