1. ## set proof problem

I need help with this problem, also what is this Z^+?

Show that A=B if A = {1,2,3} and B = {n | n ∈ Z^+ and n^2 <10}

2. ## Re: set proof problem

Originally Posted by mgk501
I need help with this problem, also what is this Z^+?

Show that A=B if A = {1,2,3} and B = {n | n ∈ Z^+ and n^2 <10}
Show that $\displaystyle A\subseteq B~\&~B\subseteq A$

3. ## Re: set proof problem

Originally Posted by mgk501
I need help with this problem, also what is this Z^+?

Show that A=B if A = {1,2,3} and B = {n | n ∈ Z^+ and n^2 <10}
$\displaystyle \displaystyle Z^{+}$ means "positive integers".

4. ## Re: set proof problem

Z^+ means the set of all positive integers
Z^+ = {1, 2, ... }

Need to show that A is a subset of B and that B is a subset of A.

Let x be in A. Then x = 1 or x = 2 or x = 3. If x = 1, then clearly x is in Z^+. Note that 1^1 = 1 < 10. Thus, x = 1 is in B. If x = 2, then clearly x is in Z^+. Note that 2^2 = 4 < 10. Then x = 2 is in B. If x = 3, then clearly x is in Z^+. Note that 3^2 = 9 < 10. Thus, x = 3 is in B. Therefore, A is a subset of B.

Now let x be in B. Then x is in Z^+ and x^2 < 10. Note that x => 1 since x is in Z^+. Note that if x > 4, then x^2 > 4^2 = 16 > 10. If x <= 3, then x^2 <= 3^2 = 9 < 10. Thus, 1 <= x <= 3. Since x is an integer we see that x = 1 or x = 2 or x = 3. Thus, x is in A. Therefore, B is a subset of A.

Therefore, A = B.