Ex 9.1, 5

Transcript

Ex 9.1, 5 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. If a and b are perfect squares, then ab is a perfect square. Let’s write Proposition and Converse first Proposition: If 𝑎 and 𝑏 are perfect squares, then 𝑎𝑏 is a perfect square. Converse: If 𝑎𝑏 is a perfect square, then 𝑎 and 𝑏 are perfect squares Checking if Proposition is true Proposition: If 𝑎 and 𝑏 are perfect squares, then 𝑎𝑏 is a perfect square. Let’s prove this Since 𝑎,𝑏 are perfect squares, write: 𝑎=𝑟^2,□( ) 𝑏=𝑠^2 for integers 𝑟,𝑠. Then: 𝑎𝑏=𝑟^2 𝑠^2=(𝑟𝑠)^2. Since 𝑟𝑠 is an integer, 𝑎𝑏 is a perfect square. Hence the proposition is true. Given 𝑥=𝑦 Adding 𝑎 both sides 𝑎+𝑥=𝑎+𝑦 Hence proved ∴ Proposition is true Checking if Converse is true Converse: If 𝑎𝑏 is a perfect square, then 𝑎 and 𝑏 are perfect squares Let’s prove this Given 𝑎+𝑥=𝑎+𝑦 Subtracting 𝑎 both sides 𝑎+𝑥−𝑎=𝑎+𝑦−𝑎 𝑥=𝑦 Hence proved ∴ Converse is true Proposition: If 𝑎 and 𝑏 are perfect squares, then 𝑎𝑏 is a perfect square. Let’s prove this Since 𝑎,𝑏 are perfect squares, We can write: 𝑎=𝑟^2,□( ) 𝑏=𝑠^2 for integers 𝑟,𝑠. Now, 𝒂𝒃=𝑟^2 × 𝑠^2 =(𝑟𝑠)^2. Since 𝑟𝑠 is an integer, 𝑎𝑏 is a perfect square. ∴ Proposition is true Checking if Converse is true Converse: If 𝑎𝑏 is a perfect square, then 𝑎 and 𝑏 are perfect squares Let’s prove this Take 𝒂=𝟐 and 𝒃=𝟖. Then: 𝑎𝑏=2 × 8 =16 =4^2 Thus, ab = 16 is a perfect square but neither 2 nor 8 is a perfect square. ∴ Converse is false ∴ Proposition is true Checking if Converse is true Converse: If 𝑎𝑏 is a perfect square, then 𝑎 and 𝑏 are perfect squares Let’s prove this Take 𝒂=𝟐 and 𝒃=𝟖. Then: 𝑎𝑏=2 × 8 =16 =𝟒^𝟐 Thus, ab = 16 is a perfect square but neither 2 nor 8 is a perfect square. ∴ Converse is false