Let $${R_1}$$ be a relation from $$A = \left\{ {1,3,5,7} \right\}$$ to $$B = \left\{ {2,4,6,8} \right\}$$ and $${R_2}$$ be another relation from $$B$$ to $$C$$ $$ = \left\{ {1,2,3,4} \right\}$$ as defined below:

i) An element $$x$$ in $$A$$ is related to an element $$y$$ in $$B$$ (under $${R_1}$$) if $$ x + y $$ is divisible by $$3$$.
ii) An element EExEE in $$B$$ is related to an elements $$y$$ in $$C$$ (under $${R_2}$$) if $$x + y$$ is even but not divisible by $$3$$.

Which is the composite relation $$R1R2$$ from $$A$$ to $$C$$?

In a class of 200 students, 125 students have taken Programming Language course, 85 students have taken Data Structures course, 65 student have taken Computer Organization coures; 50 students have taken both Programming Language and Data Structures, 35 students have taken both Programming Languages and Computer Organization; 30 students have taken both Data Structures and Computer Organization; 15 students have taken all the three course.

How many students have not taken any of the three courses?

Let A, B, C, D be $$n\,\, \times \,\,n$$ matrices, each with non-zero determination. If ABCD = I, then $${B^{ - 1}}$$ is

What values of x, y and z satisfy the following system of linear equations? $$$\left[ {\matrix{ 1 & 2 & 3 \cr 1 & 3 & 4 \cr 2 & 3 & 3 \cr } } \right]\,\,\left[ {\matrix{ x \cr y \cr z \cr } } \right]\,\, = \,\left[ {\matrix{ 6 \cr 8 \cr {12} \cr } } \right]$$$