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assignment-02-sol

# assignment-02-sol - MATH 135 Fall 2010 Solution of...

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MATH 135, Fall 2010 Solution of Assignment #2 Problem 1 . Let S = {- 5 , - 4 , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , 4 , 5 } . List all of the elements in each of the following sets. (a) A = x S x is even and x is a multiple of 3 . (b) B = x S if x is odd, then x is a multiple of 5 . (c) A B . (d) A B . Solution. (a) We have A = x S x is even x S x is a multiple of 3 = {- 4 , - 2 , 0 , 2 , 4 } ∩ {- 3 , 0 , 3 } = { 0 } (b) Since P = Q is equivalent to (NOT P ) OR Q , the statement “if x is odd, then x is a multiple of 5” is equivalent to the statement “ x is odd or x is a multiple of 5”, so we have B = x S x is even or x is a multiple of 5 = x S x is even x S x is a multiple of 5 = {- 4 , - 2 , 0 , 2 , 4 } ∪ {- 5 , 0 , 5 } = {- 5 , - 4 , - 2 , 0 , 2 , 4 , 5 } (c) Since A B , A B = B = {- 5 , - 4 , - 2 , 0 , 2 , 4 , 5 } . (d) Since A B , A B = A = { 0 } . Problem 2 . For each of the following statements, determine whether it is true when the universe of discourse is Z and whether it is true when the universe of discourse is R . (a) x y ( y 3 + x = 0) (b) x y ( y < x AND z ( z < x = z y ) ) Solution.

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