[Hernandez 1213A] Cs30 day4

What is wrong with the proof?

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CS 30

Methods of Proof

Prove the following statements: 1. Gene is taking Stat 130, which has three tests, each worth 100 points. She would like to achieve a test average of over 80. However, she is very uncomfortable with the material on the first test. Show that if Gene gets at most a 40 on the first test, then she can achieve an average of at most 80.

2. Show that there is a set A such that |℘(A)| = |A × A|.

3. Show that for all sets A, B, C, (A ∩ B)\C = (A\C) ∩ (B\C).

4. Erik is struggling and just wants to pass Physics 71. His grade is based entirely on the average of four tests, each worth 100 points. A minimum grade of 60 is required to pass. Show that if Erik is to pass the class, then at least one of his test grades must be 60 or higher.

5. Show that ∀ ∈ Z, (n3 − n) mod 3 = 0.

6. Show that ∀ ∈ Z, if 4|n, then b n+2 c = n4 .

4 √ 7. Show that 2 is irrational.

8. Show that

√ 7+ 2 5

is irrational.

9. Show that log3 5 is irrational.

10. Let x ∈ R. Show that if x is irrational, then

√

x is irrational.

Disprove the following statements: 1. For all sets A, B, C, if A ⊆ B ∪ C, then A ⊆ B and A ⊆ C. 2. Z × N = N × Z. 3. There exists an integer m ≥ 3 such that m2 − 1 is prime.

4. If p is a prime number, then 2p − 1 is also prime.