In: Advanced Math
Consider the universal context to be U = Z.
Let P(x) be the proposition 1 ≤ x ≤ 3. Let Q(x) be the proposition
∃x ∈Z, x = 2k. Let R(x) be the proposition x2 = 4. Let S(x) be the
proposition x = 1.
For each of the following statements, write out its logical
negation in symbolic notation; then, decide which claim (the
original or its negation) is True or False, and why.
(a) ∃x ∈ Z, [R(x) ∧ P (x)]
(b) ∀x ∈ Z, ∃y ∈ Z, [(S(x) ∨ Q(x)) ∧ P (y) ∧ ¬Q(y)]
(c)∃x∈Z,[S(x) ⇐⇒ (P(x)∧¬Q(x))]
When we negate a quantified statement, we negate all the quantifiers first, from left to right (keeping the same order), then we negate the statement.
Also, we have,
Hence, we solve the problems as follows:
(a) Negation of ∃x ∈ Z, [R(x) ∧ P (x)] is
The original is true because x=2 satisfies the original. That is,
(b) Negation of ∀x ∈ Z, ∃y ∈ Z, [(S(x) ∨ Q(x)) ∧ P (y) ∧ ¬Q(y)] is
The negation is true because x=5 satisfies the original. That is,
(c) Negation of ∃x∈Z,[S(x) ⇐⇒ (P(x)∧¬Q(x))] is
The original is true because x=1 satisfies the original. That is,