The empty set revisited. How does the empty set ∅ interact with subsets? Consider any set S. Is the empty set a subset of S ? Recall the definition of subset: AB⊆ precisely when every element of A is also an element of B. The empty set doesn't contain any elements, so how does it interact with this claim? If we plug ∅ and the set S into the above, we get: ∅⊆SS if every element of ∅ is an element of . 1 Look at the last part ... "if every element of ∅ is an element of S." What does this mean? Recall that there aren't any elements of ∅∅, because doesn't contain any elements! Given this, is the above statement true or false? 2 There are two ways to think about this: 1. Since ∅ contains no elements, the claim "every element of ∅ is an element of S." is false, because we can't find even a single example of an element of ∅ that is contained in S. 2. Since ∅ contains no elements, the claim "every element of ∅ is an element of S." is true, because we can't find even a single example of an element of ∅ that isn't contained in S. 3 So which line of thinking is mathematically correct? It turns out that it is the second approach, and indeed it is true that ∅⊆S. 2. Since ∅ contains no elements, the claim "every element of ∅ is an element of S." is true, because we can't find even a single example of an element of ∅ that isn't contained in S. Why is that? 4 We need to introduce the idea of a vacuous truth. Informally, a statement is vacuously true if it's true simply because it doesn't actually assert anything at all. Example: Consider the statement "if I am a cheeseburger, then the earth is flat." This statement is completely meaningless since the statement "I am a cheeseburger" is false. As a consequence, the statement "if I am a cheeseburger, then the earth is flat." doesn't actually assert anything because I'm not a cheeseburger. 5 More formally, The statement "if PQ, then " is vacuously true if P is always false. 6 More generally, The statement "Every XY has property is vacuously true if there are no X 's. 7 Back to the original question: is ∅ a subset of any set S ? Recall that ∅∅ a subset of set S if every element of is also an element of S. But the statement "every element of ∅ is an element of S " is vacuously true because there are no elements of ∅. We have For any set SS, ∅⊆ . 8 Note that since, for any set SS, ∅⊆ , we have ∅⊆{1,2,3} , ∅⊆{pig, chicken, cow} , ∅⊆ and even ∅⊆∅ Is the empty set unique? 9 Recall that two sets ST and are equal if and only if they have the same elements. This means S= T ⇔ for all xx, ∈⇔∈ S x T (1) Theorem: The empty set is unique. Proof: Suppose, BWOC, that ∅≠∅′ are empty sets. We have, by definition, if ∅ is an empty set, then ∅⊆∅′ also, if ∅′′ is an empty set, then ∅ ⊆∅ We have both ∅⊆∅′′ and ∅ ⊆∅. By (1) above we have ∅=∅′ and thus the empty set is unique. 10 Let AB and be two sets. We say that A is a subset of B, denoted by AB⊆ , if and only if every element of A is also an element of B. In symbols [ A⊆ B] ⇔[ xA ∈ ⇒∈ xB]. If there exists an element of A which is not in B we write AB⊆ . 11 Note that, for any set A, the proposition x∈∅⇒ xA ∈ is vacuously true since xA∈∅ is always false. So ∅⊆ . Two sets AB and are said to be equal if and only if BA⊆⊆ and AB. In this case we can write AB= . If two sets are not equal we write AB≠ . 12 End Presentation .