Let S= 1,2,3,4 . Then the number of elements in the set f: S × S arrow S: f is onto and f ( a , b )= f ( b , a ) ≥ a ∀( a , b ) ∈ S × S is

Question

Let S= 1,2,3,4 . Then the number of elements in the set f: S × S arrow S: f is onto and f ( a , b )= f ( b , a ) ≥ a ∀( a , b ) ∈ S × S is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

(1,1),(1,4),(4,1),(2,4),(4,2),(3,4),(4,3), (4,4)- all have one choice for image. (2,1),(1,2),(2,2)- all have three choices for image (3,2),(2,3),(3,1),(1,3),(3,3)- all have two choices for image. So the total functions =3 × 3 × 2 × 2 × 2=72 Case 1: None of the pre-images have 3 as image Total functions =2 × 2 × 1 × 1 × 1=4 Case 2: None of the pre-images have 2 as image Total functions =2 × 2 × 2 × 2 × 2=32 Case 3: None of the pre-images have either 3 or 2 as image Total functions =1 × 1 × 1 × 1 × 1=1 ∴ Total onto functions =72-4-32+1=37

Q. Let S={1,2,3,4}. Then the number of elements in the set {f:S×S→S:f is onto and f(a,b)=f(b,a) ≥a∀(a,b)∈S×S} is

Q. Let $S = {1, 2, 3, 4}$ . Then the number of elements in the set ${f : S \times S \to S : f$ is onto and $f (a, b) = f (b, a)$ $\geq a \forall (a, b) \in S \times S}$ is