Question

If A = { x : x is a natural number }, B = { x : x is an even natural number} C = {x : x is an odd natural number} and D = {x : x is a prime number }, find $A \cup C$

Answer 1

we know, if two sets A and B are given then, intersection of A and B is the set of all distinct elements that are in both A and B. it is denoted by $A\cap B$ or $B\cap A$. in figure, shaded region indicates $A\cap B$.
and A union B is the set of all distinct elements that are in Both A or B. it is denoted by $A\cup B$. in 2nd figure, it is clearly shown.

now come to the question,
given, A = {x : x is a natural number}
e.g., A = {1, 2, 3, 4, 5, 6, ...... }
B = {x : x is an even natural number }
e.g., B = {2, 4, 6, 8, 10, ....... }
C = {x : x is an odd natural number }
e.g., C = {1, 3, 5, 7, 9, ...... }
D = {x : x is a prime natural number }
e.g., D = {1, 2, 3, 5, 7, 11, 13, .....}

so, $A\cap C$ = {1, 2, 3, 4, 5, 6, ....} $\cap$ {1, 3, 5, 7, 9, ....}
= {1, 3, 5, 7, 9, ......} = C

hence, $A\cap C= C$

and $A\cup C$ = {1, 2, 3, 4, 5, 6...} U {1, 3, 5, 7, 9, 11, ...}
= {1, 2, 3, 4, 5, 6, 7, .....} = A
hence, $A\cup C=A$

Answer 2

Answer:

A U C = { 1,2,3,4,5,...} = A

Step-by-step explanation:

A = { x : x is a natural number }

=> A = { 1,2,3,4,5,6,... }

C = { x: x is an odd natural number }

=> C = { 1,3,5,7,9,...}

A U C = { x : x € A or x € C }

= { 1,2,3,4,..} U { 1,3,5,7,... }

= { 1,2,3,4,5,6,7,...}

= A

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