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Math Help - set proof problem

  1. #1
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    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}
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    Re: set proof problem

    Quote Originally Posted by mgk501 View Post
    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 A\subseteq B~\&~B\subseteq A
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  3. #3
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    Re: set proof problem

    Quote Originally Posted by mgk501 View Post
    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 Z^{+} means "positive integers".
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    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.
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