(a) Number of elements of S divisible by 3 = 300 [ number of integers from 1 to 999 , divisible by 3 is = 333. From this subtract the number of integers from 1 to 99, divisible by 3 is = 33. Hence the requisite figure is 300]. Say n( S₃) =300

Like wise n (S₄) = 224 and n(S₁₂) = 75 = n(S_3&4)

The inclusion exclusion principle is

n( S₃) U n (S₄) = n( S₃) + n (S₄) – n (S₁₂)

= 300 +224 -75

= 549

(b) It is required to find intersection of Set of 0dd numbers and the Set of numbers not divisible by 7 , between 100 to 999. The number of elements of Set S is = 900 . In Set language we have to determine S_odd ∩ S’₇ .

Now consider the number of elements of S which are odd= 450 [number of integers from 1 to 999 which are odd would be 500. From this subtract the number of integers from 1 to 99 which are odd = 50] .Number of elements of set S₇ = 128

Let’s use following terms for brevity.

S_odd=450 , odd elements of S.

S₇ = 128 . elements of S divisible by 7

S’₇ = 900-128 = 772, obviously these would be both even and odd. Out of these 772, 386 would be odd and 386 would be even.

S_odd U S’₇ = 450+386= 836

Apply the inclusion-exclusion principle

S_odd U S’₇ = S_odd + S’ ₇ - S_odd ∩ S’₇

836 = 450 + 772 – S_odd ∩ S’₇

Hence S_odd ∩ S’₇ = 450 +772 -836

= 386 Answer