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## Here's the Solution to this Question

(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

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

contrapositive:

∼ R → ∼ Q

Which translates to If P is not a rectangle, then P is not 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.

contrapositive: If n is not odd and not 2, then n is not 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.

contrapositive: If Baangloo is not Meena’s uncle or Bingli is not her aunt, then Aangloo is

not her father

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(d)

Let ,

p: A positive interger is a prime

q: Has no divisor other than 1 and itself.

contrapositive

¬q → ¬p

If a positive integer has a divisor other than 1 and itself,

then it is not prime.

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(e)

let

p:this number divisible by 3

q:this number to be divisible by 9.

contrapositive:

Since ~ p →~q is logically equivalent with q → p (its contrapositive), another valid statement would be:

If this number is divisible by 9, then this number is divisible by 3.