Class 12 Relations And Functions · Case Study4 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Case Study #197
A Ludo activity uses A={S,D} and die outcomes B={1,2,3,4,5,6}. Consider functions and relations involving these finite sets.
Part (1) 1 Marks
On B, define xRy iff x divides y. State its relation properties.
Answer: Reflexive and transitive, not symmetric.
- Each x divides itself; divisibility composes transitively, while 2 divides 4 but 4 does not divide 2.
- Hence R is reflexive and transitive but not symmetric.
Part (2) 1 Marks
How many functions A→B are possible?
Answer: 36
- Each of the two inputs has six independent choices of image, so the count is 6².
- There are 36 functions.
Part (3) 2 Marks
On B, test reflexivity, symmetry and transitivity of R={(1,2),(2,2),(1,3),(3,4),(3,1),(4,3),(5,5)}.
Answer: None of the three properties holds.
- Reflexivity fails because (1,1) is missing.
- Symmetry fails since (1,2) occurs without (2,1); transitivity fails since (1,3),(3,1) occur without (1,1).
- Thus R is neither reflexive, symmetric nor transitive.
Part (4) 2 Marks
Alternatively, how many relations from A to B are possible?
Answer: 4096
- A×B has 2·6=12 ordered pairs.
- A relation is any subset of A×B; each ordered pair is independently included or excluded, giving 2¹².
- Therefore the number of relations is 4096.