MAT2611 ASSIGNMENT 1 2025
Problem 1
(a) 𝐴 = {𝑎, 𝑏, 𝑐 }, 𝐵 = {1, 2}
𝐴 × 𝐵 = {(𝑎, 1), (𝑎, 2), (𝑏, 1), (𝑏, 2), (𝑐, 1), (𝑐, 2)}
𝐵 × 𝐴 = {(1, 𝑎) , (1, 𝑏) , (1, 𝑐 ), (2, 𝑎) , (2, 𝑏), (2, 𝑐 )}
𝐴 × 𝐵 is not equal to 𝐵 × 𝐴.
Elements in both sets are in the form (𝑥, 𝑦)
Now in order for (𝑥1 , 𝑦1 ) to be equal to (𝑥2 , 𝑦2 ) , 𝑥1 must be equal to 𝑥2 and 𝑦1 must be equal
to 𝑦2 .
Suppose that (𝑥1 ,𝑦1 ) ∈ 𝐴 × 𝐵 and (𝑥2 , 𝑦2 ) ∈ 𝐵 × 𝐴
⇒ 𝑥1 ∈ 𝐴, 𝑥2 ∈ 𝐵 and 𝑦1 ∈ 𝐵, 𝑦2 ∈ 𝐴
Since 𝐴 ≠ 𝐵, 𝑥1 can never be equal 𝑥2 and 𝑦1 can never be equal to 𝑦2
(b) |𝐴| = 𝑛, |𝐵 | = 𝑚
𝐴 = {𝑎1 , 𝑎2 , 𝑎3 , … , 𝑎𝑛 }, 𝐵 = {𝑏1 , 𝑏2 , 𝑏3 , … , 𝑏𝑚 }
Now, on 𝐴 × 𝐵, 𝑎1 will correspond to all the elements 𝑏𝑖 of B, 𝑖 ∈ {1, 2, … , 𝑚}
This will produce 𝑚 elements of 𝐴 × 𝐵
Each of the 𝑛 elements in 𝐴 will correspond to 𝑚 elements in 𝐵.
The number of elements in 𝐴 × 𝐵 is 𝑛 × 𝑚 = 𝑛𝑚
Problem 1
(a) 𝐴 = {𝑎, 𝑏, 𝑐 }, 𝐵 = {1, 2}
𝐴 × 𝐵 = {(𝑎, 1), (𝑎, 2), (𝑏, 1), (𝑏, 2), (𝑐, 1), (𝑐, 2)}
𝐵 × 𝐴 = {(1, 𝑎) , (1, 𝑏) , (1, 𝑐 ), (2, 𝑎) , (2, 𝑏), (2, 𝑐 )}
𝐴 × 𝐵 is not equal to 𝐵 × 𝐴.
Elements in both sets are in the form (𝑥, 𝑦)
Now in order for (𝑥1 , 𝑦1 ) to be equal to (𝑥2 , 𝑦2 ) , 𝑥1 must be equal to 𝑥2 and 𝑦1 must be equal
to 𝑦2 .
Suppose that (𝑥1 ,𝑦1 ) ∈ 𝐴 × 𝐵 and (𝑥2 , 𝑦2 ) ∈ 𝐵 × 𝐴
⇒ 𝑥1 ∈ 𝐴, 𝑥2 ∈ 𝐵 and 𝑦1 ∈ 𝐵, 𝑦2 ∈ 𝐴
Since 𝐴 ≠ 𝐵, 𝑥1 can never be equal 𝑥2 and 𝑦1 can never be equal to 𝑦2
(b) |𝐴| = 𝑛, |𝐵 | = 𝑚
𝐴 = {𝑎1 , 𝑎2 , 𝑎3 , … , 𝑎𝑛 }, 𝐵 = {𝑏1 , 𝑏2 , 𝑏3 , … , 𝑏𝑚 }
Now, on 𝐴 × 𝐵, 𝑎1 will correspond to all the elements 𝑏𝑖 of B, 𝑖 ∈ {1, 2, … , 𝑚}
This will produce 𝑚 elements of 𝐴 × 𝐵
Each of the 𝑛 elements in 𝐴 will correspond to 𝑚 elements in 𝐵.
The number of elements in 𝐴 × 𝐵 is 𝑛 × 𝑚 = 𝑛𝑚
