In the clap switch when we are directly supplying voltage from ac source then what is the use of battery....
wand what is the use of relay

A clap switch circuit is a classic beginner’s project.
Equipment can be switched on and off by just clapping your hands. Add a tiny microcontroller and you can easily build-in some more useful features.

The microcontroller in this circuit makes it a simple job to add some useful features that are not seen on other clap switch designs:

Changeover relay contacts enable the unit to be wired in conjunction with a manual changeover switch so that manual override of the switched equipment is always possible.
The unit is only responsive to a specific sequence of sounds i.e., two claps within a defined time window.
A safety feature masks the input for a given time window if misuse (repeated commands) is detected (useful if children have discovered how it works).
The safety feature and two-clap sequence detector can be built using TTL or CMOS flip-flops but by using a single microcontroller the circuit can be greatly simplified. A mains power supply is included so no additional power source is required.
Clap switch components list

Resistors:
R1, R6, R7 = 4kΩ7
R2 = 150kΩ
R3 = 22kΩ
R4 = 10kΩ
R5 = 150Ω
P1 = 100kΩ preset H
Capacitors:
C1 = 220μF 25V radial
C2 = 100nF
C3 = 1μF 16V
Semiconductors:
B1 = B80C1500 (round case, 80V piv, 1.5A)
D1,D2 = 1N4148
D3 = bicolour LED (red/green)
IC1 = 78L05
IC2 = PIC12F629CP, programmed
T1,T2,T3 = BC238 or BC547

Miscellaneous

JP1 = 2-way pinheader with jumper
K1 = 2-way PCB terminal block, lead pitch 7.5mm
K2 = 3- way PCB terminal block, lead pitch 7.5mm
MIC1 = 2-terminal electret microphone capsule
Re1 = bistable relay, 2 x changeover (e.g., Schrack RT314F12)
Tr1 = mains transformer 1 x 6V, min. 2VA, short-circuit proof (e.g., Marschner VN30.15/10522 or Era 030-7340.0T; Conrad Electronics # 506141)