Q: Mac air cannot connect to network

Try wireless diagnostics. Instructions here >  About Wireless Diagnostics

When you see an exclamation point in the Wi-Fi menu, first turn Wi-Fi off and back on. If it still fails to connect, then from the menu bar, select

 ▹ System Preferences... ▹ Network

Click the Assist me button and select Assistant. Follow the prompts. You may get a warning that Wi-Fi is not available, that you're too far from the base station, or that you're using the wrong password.

Assuming that you've ruled out those possibilities, restart the computer and try again. If there's no change, click the lock icon in the lower left corner of the preference pane and authenticate, if necessary. From the Location menu at the top of the window, select Edit Locations. A sheet will drop down. Click the plus-sign button to create a new location. Give it any name you want. In the new location, set up the Wi-Fi service with the same settings you used before. Click  Applyand test.

If there's still no connection, and if you have control of the router, compare its settings to Apple's recommendations. Some third-party routers may be incompatible in 802.11b/g/n radio mode. Try setting the mode to 802.11n only.