Question

Let \( A=\{1,2,3 \ldots 20\} \) let \( R_{1} \) and \( R_{2} \) two relation on A swh thet \( R_{1}=\{(a, b): b \) is divisible by \( a\} \) \( R_{2}=\{(a, b): a \) is an integral mulfiple of \( b \). Then number of elements in\( R_{1}-R_{2}\)  is equal to

Answer 1

To determine the number of elements in R₁ - R₂, let's first articulate the meaning of both relations on set A = {1,2,3,…,20}:

R₁ includes pairs (a, b) where b is divisible by a. This includes pairs like (1, 1), (1, 2),…, (1, 20) for 1; similar series for 2 up to (2, 20) (excluding odd numbers); for 3 up to (3, 18); and so on, reflecting the divisibility condition.

n(R₁) = 66

R₂ consists of pairs (a, b) where a is an integral multiple of b. Essentially, this relationship is the reverse of R₁. However, for the essence of R₁ - R₂, the key overlap comes with pairs where a = b, since those are the pairs that clearly reflect both conditions identically (i.e., a number is always divisible by itself, and it is always a multiple of itself). There are 20 such pairs corresponding to each integer in A from 1 to 20, resulting in the common elements between R₁ and R₂ (the intersection R₁ ∩ R₂) being 20 pairs.

R₁ ∩ R₂ = {(1, 1), (2, 2),…(20, 20)}

n(R₁ ∩ R₂) = 20

The difference R₁ - R₂ seeks elements present in R₁ but not in R₂. Given that R₁ and R₂ share 20 elements that are identical, to find R₁ - R₂, we subtract these 20 common elements from the total in R₁, resulting in 66 − 20 = 46 pairs.

n(R₁ - R₂) = n(R1) - n(R₁ ∩ R₂)

= n(R₁) - 20

= 66 - 20

R₁ - R₂ = 46 Pair