Mathematical Reasoning

Truth table analysis shows that (P ∨ Q) ∧ (¬P) is equivalent to Q ∧ ¬P.
Then (Q ∧ ¬P) ⇒ Q. This is a tautology.
The provided solution seems to have an error.
Let's check the options. (P ∨ Q) is a tautology. (P ∧ ¬Q) is a contradiction.
~ (P ⇒ Q) ⇔ P ∧ ¬Q is true.

A  (A -> B) -> B is a tautology

$A\to \left(B\to A\right)$

$\sim A\vee \left(B\to A\right)$

$\sim A\vee \left(\sim B\vee A\right)$

$\left(\sim A\vee B\right)\vee \left(\sim A\vee A\right)=tA\to \left(A\cup B\right)$

Statement : “If you will work, you will earn money”

contrapositive : If you will not earn money, you will not work.

Required number =  - number of solution of x₁ + x₂ + x₃ = 15, where x₁ < x₂ < x₃

For x₂ = x₁ + a, a   $\ge 1$

->3x₁ + 2a + b = 15

Coefficient of x¹⁵ in

$\left(x^3+x^6+x^9+x^{12}+x^{15}\right)$

$\left(x^1+x^2+x^3+.....+x^{10}\right)=12$

Required number = 455 – 12 = 443

As $\sim \left(A\Rightarrow B\right)=A\wedge \sim B$

$\sim \left(\left(pvr\right)\Rightarrow \left(qvr\right)\right)$

$=\left(pvr\right)\wedge \left(\sim r\wedge \sim q\right)$

$=\left(p\wedge \sim r\right)\vee \left(r\wedge \sim r\right)\wedge \left(\sim q\right)$

= $\left(p\wedge \sim r\right)$

$\left(p\wedge \left(p\to q\right)\wedge \left(q\to r\right)\right)\to r$

$\left(\left(p\wedge q\right)\wedge \left(\sim p\vee r\right)\right)\to r$

$=\sim \left(p\wedge q\wedge r\right)\vee r$

$=\sim p\vee \sim q\vee \sim r\vee r$

=> tautology

Kindly consider the following figure