Find P ({x, y, z}) − P ({x, z}), where P (S) is the power set of S (theset of all subsets of S), and S − T = {x|x ∈ S and x̸ ∈ T }.
Question
Find P({x, y, z}) − P({x, z})
where
P(S) is the power set of S (the set of all subsets of S), and
S − T = {x | x ∈ S and x̸ ∈ T}.
Solution
Sure, let's break this down:
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First, we need to understand what a power set is. The power set P(S) of a set S is the set of all possible subsets of S. For example, if S = {x, y, z}, then P(S) = { {}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z} }.
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Next, we need to understand what S - T means. This is the set of all elements in S that are not in T. For example, if S = {x, y, z} and T = {x, z}, then S - T = {y}.
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Now we can calculate P({x, y, z}) - P({x, z}). This is the set of all subsets of {x, y, z} that are not subsets of {x, z}.
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The subsets of {x, y, z} are: {}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}.
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The subsets of {x, z} are: {}, {x}, {z}, {x, z}.
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Therefore, P({x, y, z}) - P({x, z}) = { {y}, {y, z}, {x, y}, {x, y, z} }.
So, the answer is { {y}, {y, z}, {x, y}, {x, y, z} }.
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