For a given proposition there are usually44many ways to write it.Writing a good proof45is a work of art. In this chapter, you will see a46couple of common proof techniques.47Let us do a couple more examples.48Def:An integeraiseveniff∃n∈Isuch that 2n=aDef:An integeraisoddiff∃n∈Isuch that 2n+ 1 =aFor example, 6 is even since ifn= 3 then we49get 2×3 = 6. 7 is odd sincen= 3 then we get502×3 + 1 = 7.51Let us use these definitions to show that52Theorem:Assume thatxis even andyis odd53thenxyis even.54Proof:Sincexis even this means, by definition55above, that56∃n∈Isuch that 2n=xLet us call the numbernx∈I572nx=xSimilarly, forysince it is odd, by definition,58we can find an integernysuch that592ny+ 1 =yConsider the product of the two60xy= 2nx×(2ny+ 1)= 2(nx×(2ny+ 1))= 2mSincemis a product of integers,mis an integer.