Please show all work and give explanations for solutions. Thank you! Common Proof ToolsThis list serves as a reminder of proof justifications you may use throughoutthe homework. We are providing the following here as hints for yourhomework, however, this list will not be provided on an exam1. x is even if and only if x is even2. x is odd if and only if x is odd3. Multiplication is closed under integers4. Addition is closed under integers5. The definition of a biconditional states that p ↔ q is equivalent to p→q^q→p6. The list is not all inclusive and does not include other definitions learnedin class and algebraic properties learned elsewhere 3. Use a direct proof to show that if x +y is even, then x²y - y³ + 2 is even.Clearly state your reasoning for all statements and use a two-column prooffor the body whenever possible. You should include an intro, body (in twocolumn format), and a conclusion. (Hint: It might be beneficial to factorr²y - y³ in your scratch work)

Introduction: In this proof, we will show that if x+y is even, then x^2*y-y^3+2 is even. We will use…

3. Use a direct proof to show that if x +y is even, then r²y - y³ + 2 is even. Clearly state your reasoning for all statements and use a two-column proof for the body whenever possible. You should include an intro, body (in two column format), and a conclusion. (Hint: It might be beneficial to factor r²y-y³ in your scratch work)