The following system of equations
$${x_1}\, + \,{x_2}\, + 2{x_3}\, = 1$$
$${x_1}\, + \,2 {x_2}\, + 3{x_3}\, = 2$$
$${x_1}\, + \,4{x_2}\, + a{x_3}\, = 4$$ has a unique solution. The only possible value (s) for $$\alpha $$ is/are

If $$P, Q, R$$ are subsets of the universal set $$U$$, then
$$\left( {P \cap Q \cap R} \right) \cup \left( {{P^c} \cap Q \cap R} \right) \cup {Q^c} \cup {R^c}$$ is

What is the probability that in a randomly choosen group of r people at least three people have the same birthday?

Which of the following is the negation of $$$\left[ {\forall x,\alpha \to \left( {\exists y,\beta \to \left( {\forall u,\exists v,\gamma } \right)} \right)} \right]?$$$