Tag: principle of mathematical induction

$(p \wedge q) \vee  \sim p$ is equivalent to 

In a certain code language, $'543'$ means 'give my water'; $'247'$ means 'water is life' and $'632'$ means 'enjoy my life'. Which of the following stands for 'enjoy' in that language?

$\sim (p \wedge q)\Rightarrow (\sim p)\vee (\sim p \vee q)$ is equal to

The equivalent of $(p \rightarrow \sim p) \vee (\sim p \rightarrow p)$ is 

Identify which of the following statement is not equivalent to the others

Either $p$ or $q$ is equivalent to:

Equivalent statement of ''If $x\in Q$, then $x\in T$'' is
$x\in Q$ is necessary for $x\in l$
$x\in l$ is sufficient for $x\in Q$
$z\in Q$ or $x\in l$
$x\in Q$ but $x\in l$ 

Let  $P , Q , R$  and  $S$  be statements and suppose that  $P \rightarrow Q \rightarrow R \rightarrow P.$  If  $\sim S \rightarrow R,$  then

$(p\rightarrow q)\leftrightarrow (q\vee \sim p)$ is - 

Let S be a set of n persons such that:(i)any person is acquainted to exactly k other persons in s;(ii)any two persons that are acquainted have exactly $\displaystyle l $ common acquaintances in s;(iii)any two persons that are not acquainted have exactly m common acquaintances in S.Prove that $\displaystyle m\left ( n-k \right )-k\left ( k-1 \right )+k-m= 0.$