Page 85 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Equivalences Involving Quantifiers 71
As an application of the distributive law for the universal quantifier and
conjunction, suppose A and B are sets and consider the equation A = B.We
know that two sets are equal when they have exactly the same elements.
Thus, the equation A = B means ∀x(x ∈ A ↔ x ∈ B), which is equivalent
to ∀x[(x ∈ A → x ∈ B) ∧ (x ∈ B → x ∈ A)]. Because the universal quanti-
fier distributes over conjuction, this is equivalent to the formula ∀x(x ∈ A →
x ∈ B) ∧∀x(x ∈ B → x ∈ A), and by the definition of subset this means
A ⊆ B ∧ B ⊆ A. Thus, we have shown that the equation A = B is also equi-
valent to the formula A ⊆ B ∧ B ⊆ A.
We have now introduced seven basic logical symbols: the connectives ∧,
∨, ¬, →, and ↔, and the quantifiers ∀ and ∃. It is a remarkable fact that the
structureofallmathematicalstatementscanbeunderstoodusingthesesymbols,
and all mathematical reasoning can be analyzed in terms of the proper use of
these symbols. To illustrate the power of the symbols we have introduced, we
conclude this section by writing out a few more mathematical statements in
logical notation.

Example 2.2.3. Analyze the logical forms of the following statements.

1. Statements about the natural numbers. The universe of discourse is N.
(a) x is a perfect square.
(b) x is a multiple of y.
(c) x is prime.
(d) x is the smallest number that is a multiple of both y and z.
2. Statements about the real numbers. The universe of discourse is R.
(a) The identity element for addition is 0.
(b) Every real number has an additive inverse.
(c) Negative numbers don’t have square roots.
(d) Every positive number has exactly two square roots.
Solutions

1. (a) This means that x is the square of some natural number, or in other
2
words ∃y(x = y ).
(b) This means that x is equal to y times some natural number, or in other
words ∃z(x = yz).
(c) This means that x > 1, and x cannot be written as a product of two
smaller natural numbers. In symbols: x > 1 ∧¬∃y∃z(x = yz ∧ y <
x ∧ z < x).
(d) We translate this in several steps:
(i) x is a multiple of both y and z and there is no smaller number that
is a multiple of both y and z.