Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2)
First: Make sure that the owner of /var/lib/mysql and its files and subdirectories is mysql user of mysql group

to do that run the following command:

chown mysql:mysql -R /var/lib/mysql/*


Second: change write permissions to rwxr-xr-x (755) for mysql directory, and all its files and subdirectories, by running the following command:

chmod 755 -R /var/lib/mysql/*


now start mysql service 

service mysql start