Question about notation of $\bigwedge_{i=1}^{4}a_i$ and $\bigwedge_{i=1}^4 \bigvee_{j=1}^4 a_{ij}$

Question

Hi maths people I have question about notations

$$\bigwedge_{i=1}^{4}a_i \text{ }\text{ and }\text{ } \bigwedge_{i=1}^4 \bigvee_{j=1}^4 a_{ij}$$

I think first one $\bigwedge_{i=1}^{4}a_i$ I understanded. You can write it like this:

$$\bigwedge_{i=1}^{4}a_i= a_1 \wedge a_2 \wedge a_3 \wedge a_4$$

But how is writed correct $\bigwedge_{i=1}^4 \bigvee_{j=1}^4 a_{ij}$ ?

Answer 1

$$\bigwedge_{i=1}^4 \bigvee_{j=1}^4 a_{ij} =$$

$$ (a_{11} \lor a_{12} \lor a_{13} \lor a_{14}) \land (a_{21} \lor a_{22} \lor a_{23} \lor a_{24}) \land (a_{31} \lor a_{32} \lor a_{33} \lor a_{34}) \land (a_{41} \lor a_{42} \lor a_{43} \lor a_{44})$$

Answer 2

The third $i$ (in $a_{i}$) is bounded by the second $i$ ($\vee$), and the first $i$ ($\wedge$) bounds nothing, so $\wedge_{i}\vee_{i}a_{i}=\vee_{i}a_{i}$.