simple intersection understanding problem

Question

I am reading theoretical computer science subject and came across this small step.

in the textbook, they defined two contextfree languages:

$L_1 = \{ a^i b^j c^j | i,j>0\}$ and $L_2 = \{a^i b^i c^j | i,j>0\}$

and they say, the intersection of these two languages are not contextfree anymore. because the Intersection is

$$L_1 \cap L_2 = \{ a^ib^ic^i | i>0\}$$

but why is it $a^ib^ic^i$ ? , i thought, what these two language have in common is only $a^i$.

please guide my a bit

many thanks

Answer 1

Look at these definitions of L1 and L2:

L1 = {The expressions in form of $a^xb^yc^z$ where y=z}

L2 = {The expression in form of $a^xb^yc^z$ where x=y}

So the intersection is {The expression in form of $a^xb^yc^z$ where x=y AND y=z}

Therefore, $a^ib^ic^i$