1. Given 10 numbers a1, a2,..., a10, from the set S of natural numbers S = {1,2,3,..., 100}. Prove that there exist two (or more) subsets of the 10 numbers whose elements sum to the same value. Food for thought - how would you determine that 10 values are needed? 2. Given the set of S natural numbers S= {1, 2, 3,..., 100} select fifty one numbers from S. Prove that at least one of the numbers you chose is a multiple of another number that you chose. 3. Evan is training for baseball season. His coach wants him to pitch or hit 90 times over the course of the 53 days before the season starts. He plans to do at least one pitching or hitting session a day. Prove that there exists a span of consecutive days that Evan will complete exactly 15 sessions. 4. Use the following Venn diagram for this question. 3 (a) List the regions of the Venn diagram that make the following statement true: (-p→q) → r. (b) Construct an equivalent statement to the one in (a) using only A, V, and/or using as few connectives as possible. (c) List the regions of the Venn diagram that make the following statement true: p→ (q→→r). (d) Is→ commutative? associative? Explain your answer.
1. Given 10 numbers a1, a2,..., a10, from the set S of natural numbers S = {1,2,3,..., 100}. Prove that there exist two (or more) subsets of the 10 numbers whose elements sum to the same value. Food for thought - how would you determine that 10 values are needed? 2. Given the set of S natural numbers S= {1, 2, 3,..., 100} select fifty one numbers from S. Prove that at least one of the numbers you chose is a multiple of another number that you chose. 3. Evan is training for baseball season. His coach wants him to pitch or hit 90 times over the course of the 53 days before the season starts. He plans to do at least one pitching or hitting session a day. Prove that there exists a span of consecutive days that Evan will complete exactly 15 sessions. 4. Use the following Venn diagram for this question. 3 (a) List the regions of the Venn diagram that make the following statement true: (-p→q) → r. (b) Construct an equivalent statement to the one in (a) using only A, V, and/or using as few connectives as possible. (c) List the regions of the Venn diagram that make the following statement true: p→ (q→→r). (d) Is→ commutative? associative? Explain your 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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Transcribed Image Text:1. Given 10 numbers a1, a2,..., a10, from the set S of natural numbers S = {1,2,3,..., 100}. Prove that there exist two (or
more) subsets of the 10 numbers whose elements sum to the same value. Food for thought - how would you determine
that 10 values are needed?
2. Given the set of S natural numbers S= {1, 2, 3,..., 100} select fifty one numbers from S. Prove that at least one of the
numbers you chose is a multiple of another number that you chose.
3. Evan is training for baseball season. His coach wants him to pitch or hit 90 times over the course of the 53 days before the
season starts. He plans to do at least one pitching or hitting session a day. Prove that there exists a span of consecutive days
that Evan will complete exactly 15 sessions.
4. Use the following Venn diagram for this question.
3
(a) List the regions of the Venn diagram that make the following statement true: (-p→q) → r.
(b) Construct an equivalent statement to the one in (a) using only A, V, and/or using as few connectives as possible.
(c) List the regions of the Venn diagram that make the following statement true: p→ (q→→r).
(d) Is→ commutative? associative? Explain your answer.
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