Ex 9.1, 9

Transcript

Ex 9.1, 9 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 8–12, n is a positive integer If n is divisible by 60, then it is divisible by both 5 and 12. Let’s write Proposition and Converse first Proposition: If n is divisible by 60, then it is divisible by both 5 and 12 Converse: If n is divisible by both 5 and 12, then it is divisible by 60 Checking if Proposition is true Proposition: If n is divisible by 60, then it is divisible by both 5 and 12 Let’s prove this Since n is divisible by 60, we can write 𝒏=𝟔𝟎𝒌 for an integer 𝑘. Now, we can write 𝑛=60𝑘=𝟓 × (𝟏𝟐𝒌) This means n is divisible by 5 And: 𝑛=60𝑘=𝟏𝟐 × (𝟔𝒌) This means n is divisible by 12 Thus 𝑛 is divisible by both 5 and 12 ∴ Proposition is true Checking if Converse is true Converse: If n is divisible by both 5 and 12, then it is divisible by 60. Let’s prove this Since n is divisible by 5, we can write n = 5p And n is divisible by 12, we can write n = 12q n = 2 × 2 × 3 × q Now, for n to be disable by both 5 and 12, its prime factorisation should be like n = 5 × 2 × 2 × 3 × k n = 60k Since n = 60k for any integer k ∴ Converse is true Let’s prove this Since n is divisible by 60, we can write 𝒏=𝟔𝟎𝒌 for an integer 𝑘. Now, we can write 𝑛=60𝑘=𝟓 × (𝟏𝟐𝒌) This means n is divisible by 5 And: 𝑛=60𝑘=𝟏𝟐 × (𝟔𝒌) This means n is divisible by 12 Thus 𝑛 is divisible by both 5 and 12 ∴ Proposition is true Checking if Converse is true Converse: If n is divisible by both 5 and 12, then it is divisible by 60. Let’s prove this Since n is divisible by 5, we can write n = 5p And n is divisible by 12, we can write n = 12q n = 2 × 2 × 3 × q Now, for n to be disable by both 5 and 12, its prime factorisation should be like n = 5 × 2 × 2 × 3 × k n = 60k Since n = 60k for any integer k ∴ Converse is true Thus 𝑛 is divisible by both 5 and 12 ∴ Proposition is true Checking if Converse is true Converse: If n is divisible by both 5 and 12, then it is divisible by 60. Let’s prove this Since n is divisible by 5, we can write n = 5p And n is divisible by 12, we can write n = 12q n = 2 × 2 × 3 × q Now, for n to be disable by both 5 and 12, its prime factorisation should be like n = 5 × 2 × 2 × 3 × k n = 60k Since n = 60k for any integer k ∴ Converse is true Now, for n to be divisible by both 5 and 12, its prime factorisation should be like n = 5 × 2 × 2 × 3 × k n = 60k Since n = 60k for any integer k It is divisible by 60 ∴ Converse is true