Q. 18 Let be the Signum Function defined as
and be the Greatest Integer Function given by , where is greatest integer less than or equal to . Then, does and coincide in ?
Q.16 Let . Then number of relations containing and which are
reflexive and symmetric but not transitive is
Q. 15 Let , and be functions defined by and . Are and equal?
Justify your answer. (Hint: One may note that two functions and
such that , are called equal functions).
Q. 14 Define a binary operation ∗ on the set as
Show that zero is the identity for this operation and each element of the set
is invertible with being the inverse of .
Q. 13 Given a non-empty set X, let be defined as
Show that the empty set is the
identity for the operation ∗ and all the elements A of P(X) are invertible with
. (Hint : and ).
Q 13 ) Given a non-empty set X, let ∗ : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set φ is the identity for the operation ∗ and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – φ) ∪ (φ – A) = A and (A – A) ∪ (A – A) = A ∗ A = φ).
Q. 12 Consider the binary operations and defined as
and . Show that ∗ is commutative but not
associative, is associative but not commutative. Further, show that ,
. [If it is so, we say that the operation ∗ distributes
over the operation ]. Does distribute over ∗? Justify your answer.
Q. 9 Given a non-empty set X, consider the binary operation
given by , where P(X) is the power set of X.
Show that X is the identity element for this operation and X is the only invertible
element in P(X) with respect to the operation ∗.
Q.8 Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows:
For subsets A, B in P(X), ARB if and only if . Is R an equivalence relation
on P(X)? Justify your answer.
Q. 6 Give examples of two functions and such that is
injective but g is not injective.
(Hint : Consider and ).
Q. 2 Let be defined as , if n is odd and , if n is
even. Show that f is invertible. Find the inverse of f. Here, W is the set of all