DHARMDISCRETE THEORY, RELATIONS AND FUNCTIONS 21
1.2 Define the finite set and infinite set; countable set and uncountable set. State which set is finite
or infinite, countable or uncountable?
(i) {.........2BC, 1BC, 1AD, 2AD, ........2004AD, .......}
(ii) {0, 1, 2, 3, ..........}
(iii){x/x is positive integer}
(iv){x/x ∈ {a, b, c, ...... y, z}}
(v) The set of living beings on the universe.
(vi) The set of lines passes through the origin.
(vii)X = {x/x^2 + 1 = 0}
(viii) X = {1/2, 3, 5, 2 , 7, 9}.
1.3 Does every set have a proper subset?
1.4 Prove that if A is the subset of ∅ then A = ∅.
1.5 Find the power set of the set X = {a, {a, b}, {∅}}.
1.6 Prove if X ∩ Y = ∅ then X ⊂ Y′.
1.7 Consider the universal set U = {x/x is a integer}, and the set X = {x/x is a positive integer}, Z =
{x/x is a even integer}, and Y = {x / x is a negative odd integer}, then find,
(i)X ∪ Y(ii)X′ ∪ Y
(iii) X – Y (iv)Z – Y′
(v) (X ∩ Y) – Z′
1.8 Define a binary relation. When a relation is said to be reflexive, symmetric, and transitive.
1.9 Distinguish between a relation and a mapping.
1.10 Let a relation R = {(x, y)/x, y? R and 4x^2 + 9y^2 = 36} then find the domain of R, the range of R, and
the R–1.
1.11 State the condition when a relation R in a set X
(i) not reflexive (ii) not symmetric
(iii) not antisymmetric (iv) not transitive.
1.12 Let R 1 and R 2 are two relations then in a set X then prove the following :
(i) If R 1 and R 2 is symmetric then R 1 ∪ R 2 is also symmetric.
(ii) If R 1 is reflexive then R 1 ∩ R 2 is also reflexive.
1.13 Let a relation R = {(x, y)/x, y ∈ N and (x – y) is divisible by 3}then Show that R is an equivalence
relation.
1.14 Comment on the relation R i.e., if R ∩ R–1 = ∅ and if R = R–1.
1.15 Let X be the set of people and R be the relation defined between the element of the set X, i.e.
(i) R = {(x, y)/x, y ∈ X and ‘x is the husband of y’}
(ii) R = {(x, y)/x, y ∈ R and ‘x is poorer than y’}
(iii) R = {(x, y)/x, y ∈ R and ‘x is younger than y’}
(iv) R = {(x, y)/x, y ∈ R and ‘x is thirsty than y’}
Find the inverse of each of the relation.
1.16 If X = {1, 2}, Y = {a, b, c} and Z = {c, d}. Find (A × B) ∩ (A × C).