1) How do I prove the following: Let A = { 6 a + 4...

Aphroditeoq

Aphroditeoq

Answered

2022-07-15

1) How do I prove the following: Let A = { 6 a + 4 b Z : a , b Z } and B = { 2 a Z : a Z }. Show that A = B.

Answer & Explanation

Brendon Bentley

Brendon Bentley

Expert

2022-07-16Added 11 answers

Explanation:
You prove it by proveing that every number that can be written as 6 a + 4 b where a,b are integers can be written as 2m where m is an integer and vice versa.
If x A then x = 6 a + 4 b for some a , b Z .
so..... you prove that there is an m Z so that..... x = 2 m.
Step 2
So x B. So A B.
Then if y B then x = 2 n for some n Z .
so .... you prove that there are a , b Z so that ... y = 6 a + 4 b.
So y A. So B A.
So A B and B A so A = B.
ganolrifv9

ganolrifv9

Expert

2022-07-17Added 4 answers

Step 1
First direction: show that A B.
Let x A, then x = 6 a + 4 b for some x B
Since x = 2 ( 3 a + 2 b ), it is a multiple of two and in particular, x B.
Step 2
Second direction: show that B A
Conversely, let x B, then x = 2 a for some a Z .
Then x = 6 a + 4 ( a ), so x A. (You can pick anything in the brackets)
Hence A = B

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?