Page 54 - HOW TO PROVE IT: A Structured Approach, Second Edition

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40 Sentential Logic
You can now check, using equivalences from Section 1.2, that these statements
are equivalent. An alternative way to check the equivalence is with a truth table.
To simplify the truth table, let’s use P and Q as abbreviations for the statements
x ∈ A and x ∈ B. Then we must check that the formulas (P ∨ Q) ∧¬(P ∧ Q)
and (P ∧¬Q) ∨ (Q ∧¬P) are equivalent. The truth table in Figure 7 shows
this.

Figure 7

Definition 1.4.5. Suppose A and B are sets. We will say that A is a subset of
B if every element of A is also an element of B. We write A ⊆ B to mean that
A is a subset of B. A and B are said to be disjoint if they have no elements in
common. Note that this is the same as saying that the set of elements they have
in common is the empty set, or in other words A ∩ B = ∅.

Example 1.4.6. Suppose A = {red, green}, B = {red, yellow, green, purple},
and C = {blue, purple}. Then the two elements of A, red and green, are both
also in B, and therefore A ⊆ B. Also, A ∩ C = ∅,so A and C are disjoint.

If we know that A ⊆ B, or that A and B are disjoint, then we might draw a
Venn diagram for A and B differently to reflect this. Figures 8 and 9 illustrate
this.

Figure 8 Figure 9

Justaswe earlier derivedidentities showingthat certain setsare alwaysequal,
it is also sometimes possible to show that certain sets are always disjoint, or
that one set is always a subset of another. For example, you can see in a Venn