1. If A is a subset of B, then every element of set A is also an element of set
ID: 3727182 • Letter: 1
Question
1. If A is a subset of B, then every element of set A is also an element of set B.
2. There are no elements in the empty set.
3. If A is a proper subset of B, then A is a subset of B but there is at least one element of set B that is not an element of set A.
Translate "the empty set is not a proper subset of every set" into a predicate logic expression and then use 2 of the rules from question 3 above (along with the inference rules and the logical equivalences) to prove that claim. You may use any of the proof techniques described in class, but your proof must include at least 2 of the rules above
Explanation / Answer
Given a set X, a proper subset is any set Y such that YX and YX; that is, Y is contained in X but is not equal to X. This is denoted by YX in some texts.
So while X is a subset of itself, it is not a proper subset of itself. And this is true for any set, even the empty set (or void set).
Speaking of which, the empty set is not only a subset of any set, but also a proper subset of any non-empty set.
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