Solved Homework 1 CSCI 301 1. [2 points] Explicitly write out the contents of the following set: 𝑋𝜖 𝒫({1,2,3}): 2𝜖𝑋

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Homework 1
CSCI 301
1. [2 points] Explicitly write out the contents of the following set:
𝑋𝜖 𝒫({1,2,3}): 2𝜖𝑋
2. [3 points] Negate the following statement:
If x is a rational number and 𝑥 ≠ 0, then tan(x) is not a rational number.
Prove each of the following statements.
Explicitly state which method of proof you are using: direct, contrapositive, contradiction
etc.
3. [5 points] Suppose 𝑎 𝜖ℤ. If a2 is not divisible by 4, then a is odd.
4. [5 points] Suppose, 𝑎, 𝑏 𝜖 ℤ. If 4 | (𝑎7 + 𝑏7), then a and b are not both odd.
5. [10 points] Given an integer a and b, then a2
(b+3) is even if and only if a is even or b is
odd.