Ex 9.1, 4

Transcript

Ex 9.1, 4 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 x = y, then a + x = a + y, where x, y and a are any three numbers. This proposition and its converse are routinely used while solving equations. Let’s write Proposition and Converse first Proposition: If 𝑥=𝑦, then 𝑎+𝑥=𝑎+𝑦, where 𝑥,𝑦 and 𝑎 are any three numbers. Converse: If 𝑎+𝑥=𝑎+𝑦, then 𝑥=𝑦, where 𝑥,𝑦 and 𝑎 are any three numbers. Checking if Proposition is true Proposition: If 𝑥=𝑦, then 𝑎+𝑥=𝑎+𝑦, where 𝑥,𝑦 and 𝑎 are any three numbers. Let’s prove this Given 𝑥=𝑦 Adding 𝑎 both sides 𝑎+𝑥=𝑎+𝑦 Hence proved ∴ Proposition is true Checking if Converse is true Converse: If 𝑎+𝑥=𝑎+𝑦, then 𝑥=𝑦, where 𝑥,𝑦 and 𝑎 are any three numbers. 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 Checking if Proposition is true Proposition: If 𝑥=𝑦, then 𝑎+𝑥=𝑎+𝑦, where 𝑥,𝑦 and 𝑎 are any three numbers. Let’s prove this Given 𝑥=𝑦 Adding 𝑎 both sides 𝑎+𝑥=𝑎+𝑦 Hence proved ∴ Proposition is true Checking if Converse is true Converse: If 𝑎+𝑥=𝑎+𝑦, then 𝑥=𝑦, where 𝑥,𝑦 and 𝑎 are any three numbers. 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 Checking if Converse is true Converse: If 𝑎+𝑥=𝑎+𝑦, then 𝑥=𝑦, where 𝑥,𝑦 and 𝑎 are any three numbers. 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