Discrete Mathematics
Assignment-1(Set Theory)
1. At a hotel, 93 people preferred coffee as a beverage, 43 people preffered juice, 25 preferred both coffee and juice. If each person prefers at least one of the coffee and juice then how many people visited the hotel.
2. In a class of 50 students ,30 speak Hindi ,15 speak both Hindi and English. How many students speak in English.
3.Show that [(A ∪ B' ∪ C') ∩ (A ' ∪ B' ∪ C' )] = B ∩ C
4.Show that (A - C) ∪ (B - C) = (A ∪ B) - C
5.Show that (A ∪ C) ∩ (A ∪ B') ∩ (A' ∪ B) = A ∩ B
6.Show that (A ∩ B) 
(A ∩ C) = A ∩( B C)
7.Show that (A ∪ B) ∩ (B ∪ C) ∩ (C ∪ A) = (A ∩ B) ∪ (B ∩ C) ∪ (C ∩ A)
8.Calculate (A - B) ∩ (B - A)
9. Let A,B,C be subsets of a universal set S.Prove that
a) Ax (B ∩ C) = (AxB)∩(AxC)
b) Ax (B ∪ C) = (AxB)∪(AxC)
c) Ax(B - C) = (AxB) - (AxC)
Relation
1. Let ρ be a relation on a set Z by aρb if and only if ab>0 for all a,b in Z. Check ρ is reflective,symmetry and transitive.
2. a) Let ρ = { (x, y)∈ ZxZ such that x.y ≥ x
b) Let ρ = { (x, y)∈ ZxZ such that x-y is multiple of 5 }
Check ρ is reflective ,symmetric ,transitive and hence an equivalence relation.
Mapping
1. Let f:A→B , g:B→C, h:B→C be three mapping such that f is surjective and g o f = h o f
Prove that g = h
2. If f:X→Y is a mapping and A,B are subsets of X ,then f(A ∩ B) = f(A) ∩ f(B)
3. If f:A→B be invertible, then f-1:B A is also invertible.
4. If f:A→B and g:B→C be two bijective mapping, prove that g o f: A → C is also bijective.