Mathematical Reasoning

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

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

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

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

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

$\mathrm{(A})\sim \left(p\wedge \left(\sim r\right)\right)\vee q=\left(\sim p\right)\vee \left(r\right)\vee q$

(B) $q\vee \left(-r\vee p\right)$

(C) $\left(\sim p\right)\vee \left(q\vee r\right)$

(D) $\left(\sim p\vee q\right)\vee r$

Using Venn diagram we get B as the correct option.

P : Ramu is intelligent

Q : Ramu is rich

R : Ramu is not honest

Give statement, “Ramu is intelligent and honest if any only if Ramu is not rich”

$=\left(P\wedge \sim R\right)\iff \sim Q$

So, negation of the statement is

$\sim \left[\left(P\wedge \sim R\right)\Rightarrow \sim Q\right]$

$=\left(\left(P\wedge \sim R\right)\wedge Q\right)\vee \left(\sim Q\wedge \left(\sim P\vee R\right)\right)$