Class 12 Relations And Functions · Case Study4 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Case Study #199
A school activity selects boys B={b₁,b₂,b₃} and girls G={g₁,g₂}. Functions and relations are formed using these sets.
Part (1) 1 Marks
How many relations from B to G are possible?
Answer: 64
- There are 3·2=6 ordered pairs in B×G; any of its subsets is a relation.
- The number of relations is 2⁶=64.
Part (2) 1 Marks
How many functions from B to G are possible?
Answer: 8
- Each of three boys can be assigned either of two girls, independently.
- The number of functions is 2³=8.
Part (3) 2 Marks
On B define xRy iff x and y have the same sex. Is R an equivalence relation?
Answer: Yes; R=B×B.
- All members of B are boys, so every ordered pair in B×B satisfies the predicate.
- The universal relation contains every diagonal pair and reversal and every required composed pair.
- Thus R is reflexive, symmetric and transitive, hence an equivalence relation.
Part (4) 2 Marks
Alternatively, is f={(b₁,g₁),(b₂,g₂),(b₃,g₁)} a bijection B→G?
Answer: No; onto but not one-one.
- The distinct inputs b₁,b₃ both have image g₁, so f is not injective.
- Both g₁ and g₂ occur as images, so f is onto G.
- Therefore f is onto but not one-one and is not bijective.