Question

Let A = {x : x $\in \mathbb{N}$}, B = {x : x = 2n, n$\in \mathbb{N}$}, C = {x : x = 2n – 1, n$\in \mathbb{N}$} and D = {x : x is a prime natural number}. Find A $\cap $ C.

Answer 1

Hint: As we are the given sets A, B, C and D in terms of n. So, we will apply substitution here. In the place of n we will substitute those values of n which satisfies the definition of the sets individually. Also we will apply intersection which means the collection between common elements of sets.

Complete step-by-step answer:
Now we will consider the sets one by one and elaborate them according to their definition. First we will consider the set A = {x : x $\in \mathbb{N}$}. Here we can clearly see that this set contains all elements of natural numbers. Therefore we have A = {1, 2, 3, 4, 5, ...}.
Now we will consider B = {x : x = 2n, n$\in \mathbb{N}$}. Here if we substitute n = 1 we get x = 2. Similarly, we get the elements as a multiple of 2. Therefore, we have B = {2, 4, 6, 8, ...}.
Now we will consider the well defined set C = {x : x = 2n – 1, n$\in \mathbb{N}$}. After substituting n = 1 we have x = 2(1) – 1 or, x = 1. And solving in this manner we will have the C = {1, 3, 5, 7, ...}.
Now we will consider the last set which is D = {x : x is a prime natural number}. It clearly says that the set D is the collection of all prime numbers. Therefore, we have that D = { 2, 3, 5, 7, ...}.
Now to get the value of the expression A $\cap $ C we see that we will take the common sets between the elements of sets A and C. Thus, we have A $\cap $ C = {1, 2, 3, 4, 5, ...} $\cap $ {1, 3, 5, 7, ...}. By applying the intersection between these two sets we get the common elements as, A $\cap $ C = {1, 3, 5, 7, ...}.
Hence, A $\cap $ C = {1, 3, 5, 7, ...} or C only.

Note: Sometimes the intersection between the two sets can be one of those sets also. As in this solution we have the intersection A $\cap $ C = C only. But this cannot be always the case. The symbol of dots after elements in the sets means that these sets are never ending. So, we also should not write these sets till five terms or any number of terms. These should be written as infinite terms.