Answer to Question #173518 in Discrete Mathematics for ANJU JAYACHANDRAN

Question

2 d) Write down the converse of each of the following statements: (2)

i) If p is a prime number and a and b are any two natural numbers and if p

divides a or b, then p divides ab.

ii) In a triangle 4ABC, if AB2 +AC2 = BC2

, then ∠BAC = 90◦

.

Answer 1

i) Let us denote statements:

A -  p is a prime number 

B - a and b are any two natural numbers

C -  p divides a or b

D - p divides ab

Then this statement has the form:

$A \wedge B \wedge C \to D$

Let us construct the converse statement:

$D \to A \wedge B \wedge C$

That is, we have the statement:

"If p divides ab then  p is a prime number and a and b are any two natural numbers and p

divides a or b"

Answer: If p divides ab then  p is a prime number and a and b are any two natural numbers and p

divides a or b

ii)

Let us denote statements:

E -  In a triangle ABC $A{B^2} + A{C^2} = B{C^2}$

F - ∠BAC = 90◦

Then this statement has the form:

$E \to F$

Let us construct the converse statement:

$F \to E$

That is, we have the statement:

"If In a triangle ABC ∠BAC = 90◦ then $A{B^2} + A{C^2} = B{C^2}$ ".

Answer: If In a triangle ABC ∠BAC = 90◦ then $A{B^2} + A{C^2} = B{C^2}$