Answer to Question #203961 in Discrete Mathematics for Talha

Question

Write CONVERSE of the following statements.

a)If P is a square, then P is a rectangle.

b)If n is prime, then n is odd or n is 2.

c)If Aangloo is Meena's father, then Baangloo is her uncle and Bingli is her Aunt.

d)A positive integer is prime only if it has no divisors other than 1 and itself.

e)Being divisible by 3 is a necessary condition for this number to be divisible by 9.

Answer 1

(a) If P is a square, then P is a rectangle.

Let Q be “P is a square” and R be “P is a rectangle.” Then we have:

Original: Q → R

=∼ Q ∨ R

Negation: ∼ (∼ Q ∨ R)

= Q ∧ ∼ R

Which translates to P is a square and not a rectangle.

Converse:

R → Q

Which translates to If P is a rectangle, then P is a square.

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(b) If n is prime, then n is odd or n is 2. Let Q be “n is prime” and R be “n is odd or n is 2.”

Then we have:

Original: Q → R =∼Q∨R

Negation: ∼ (∼ Q∨R) =Q∧∼R

If we negate R, we have:

∼(n is odd∨n is 2) =∼(n is odd) ∧ ∼ (n is 2)

Which translates to n is even and n is not 2.

Converse: If n is odd or n is 2, then n is prime.

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(c) If aangloo is meena’s father, then Baangloo is her uncle and bingli is her aunt.

Let Q be “aangloo is meena's father” and R be “baangloo is her uncle and bingli is her aunt.”

Then, as before

∼(Q→R)=Q∧∼R

If we negate R, we have:

∼(baangloo is her uncle ∧bingli is her aunt) ∼(baangloo is her uncle) ∨ ∼ (bingli is her aunt)

Which translates to baangloo is not her uncle or bingli is not her aunt. Negation: aangloo is meena's father and either baangloo is not her uncle or bingli is not her aunt.

Converse: If baangloo is her uncle and bingli is her aunt, then aangloo is meena's father.

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(d)

Let ,

p: A positive interger is a prime

q: Has no divisor other than 1 and itself.

Converse

q → p

A positive integer is a prime if it has no divisors other than 1 and itself.

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(e)

let

p:this number divisible by 3

q:this number to be divisible by 9.

Converse

q → p

this number divisible by 3 if this number to be divisible by 9.