Codibg languagr LATEX please provide input code

1. Prove that 2 is not a rational number. (a) Write down a description of the rational numbers Q in set-builder notation. (b) Write down a definition of 2. (c) Use (a) and (b) to derive a contradiction proving that 2 cannot be rational. 2. Use the Division Algorithm to prove that every odd integer is either of the form 4k+1 or 4k+3 for some kZ. 3. Which of the following sets are nonempty? Explain your answer for each. (a) A={rQ:r2=2} (b) B={rR:r2+5r7=0} (c) C={tZ:6t2t1=0}