Ex 9.1, 7

Transcript

Ex 9.1, 7 Frame the converse for each of the propositions in Questions 1–12. Then, determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement. In Questions 6 and 7, x and y are real numbers. If x = y, then x³ = y³. Let’s write Proposition and Converse first Proposition: If 𝑥=𝑦, then 𝑥^3=𝑦^3 Converse: If 𝑥^3=𝑦^3, then 𝑥=𝑦 Checking if Proposition is true Proposition: If 𝑥=𝑦, then 𝑥^2=𝑦^2 Let’s prove this Given 𝑥=𝑦 Cubing both sides 𝑥^3=𝑦^3 Hence proved ∴ Proposition is true Checking if Converse is true Converse: If 𝑥^3=𝑦^3, then 𝑥=𝑦 Let’s prove this Let 𝑥 = 2, 𝑦 = –2 Now, 𝑥^2=2^2=4 𝑦^2=〖(−2)〗^2=4 Thus, 𝑥^2=𝑦^2 but 𝒙≠𝒚 ∴ Converse is false Proposition: If 𝑥=𝑦, then 𝑥^2=𝑦^2 Let’s prove this Given 𝑥=𝑦 Cubing both sides 𝑥^3=𝑦^3 Hence proved ∴ Proposition is true Checking if Converse is true Converse: If 𝑥^3=𝑦^3, then 𝑥=𝑦 Let’s prove this Since 𝑥^3=𝑦^3 𝒙^𝟑−𝒚^𝟑=𝟎 (𝑥−𝑦)(𝑥^2+𝑦^2+𝑥𝑦) =0 Thus (𝑥−𝑦)=0 𝒙=𝒚 𝑥^2+𝑥𝑦+𝑦^2 =0 (𝒙+𝒚/𝟐)^𝟐+(𝟑𝒚^𝟐)/𝟒=𝟎 Since terms on the left is sum of squares It can be 0 only if x = 0, y = 0 In either case: 𝑥=𝑦". " ∴ Converse is true