Class 12 Relations And Functions · Case Study5 Marksmixed_relations_and_functions
Class 12 Maths · Relations And Functions · Case Study #64
Ludo: relations and functions
Part (1) 1 Marks
On \(B\), define \(R=\{(x,y)\in B\times B:x\mid y\}\), so \(y\) is divisible by \(x\). The relation is
Answer: Option (A): Reflexive and transitive but not symmetric
- Every element divides itself, so the relation is reflexive. Divisibility is transitive. It is not symmetric, since \(1\mid2\) but \(2\nmid1\).
Part (2) 1 Marks
Raji wants to know the number of functions from \(A\) to \(B\). How many functions are possible?
Answer: Option (A): \(6^2\)
- Each of the two domain elements has six choices of image: \(6^2\).
Part (3) 1 Marks
Let \(R\) be a relation on \(B\) defined by \(R=\{(1,2),(2,2),(1,3),(3,4),(3,1),(4,3),(5,5)\}\). Then \(R\) is
Answer: Option (D): None of these
- Reflexivity fails because \((1,1)\) is absent; symmetry fails because \((2,1)\) is absent; transitivity fails because \(1R3\) and \(3R4\), but not \(1R4\).
Part (4) 1 Marks
Raji wants to know the number of relations possible from \(A\) to \(B\). How many are possible?
Answer: Option (D): \(2^{12}\)
- \(A\times B\) has \(2\cdot6=12\) pairs, each independently included or omitted. Thus there are \(2^{12}\) relations.
Part (5) 1 Marks
On \(B\), let \(R=\{(1,1),(1,2),(2,2),(3,3),(4,4),(5,5),(6,6)\}\). Then \(R\) is
Answer: Option (B): Reflexive and transitive
- All diagonal pairs are present and the only non-diagonal pair is \((1,2)\), which introduces no missing transitive consequence. Its reverse is absent.