so for a homework assignment i need to prove the following:

We have arbitrary languages L1⊆∑1*, L2⊆∑2*, L3⊆∑3*, L4⊆∑4*

Prove that the followging is either true or false:

If L1 ∪ L2 ⊆ L3 ∩ L4, then L1◦L2 ⊆ L3◦L4 (◦ = concatenation)

Now i know this is true, but i don't know how to approach this problem...

Any tips on pointing me into the right direction are appreciated!


1 Answer 1


The assignment tests basic definitions (and some elementary set theory).

As you know, for arbitrary $L_1,L_2$, you have $L_1\cdot L_2 = \{ x\cdot y \mid x\in L_1, y\in L_2\}$.

Now if $L_1\subseteq L_3$ and $L_2\subseteq L_4$ then $L_1\cdot L_2 \subseteq L_3\cdot L_4$. This is rather clear, or if you want, can shown with elementary steps. Take any $z\in L_1\cdot L_2$, then $z=xy$ for $x\in L_1$ and $y\in L_2$. However, because of the inclusions, $z=xy$ for $x\in L_3$ and $y\in L_4$, thus $z\in L_3 \cdot L_4$.

  • $\begingroup$ Yeah i get this, but the only information i have is --> L1 ∪ L2 ⊆ L3 ∩ L4 How would i use this for showing that e.g. L1⊆L3 and L2⊆L4 $\endgroup$
    – GSerum_
    Commented Apr 30, 2018 at 15:14
  • $\begingroup$ That is basic set theory, and is not particular tolanguages. For example $A\subseteq B\cap C$ if and only if $A\subseteq B$ and $A\subseteq C$. $\endgroup$ Commented May 1, 2018 at 23:38

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