Question

Write down the negation of each statement

1. ∃y ∈Z such that ∀x ∈Z, R (x + y)

2. ∀x ∈Z, ∃y∈Z such that R(x + y)

Answer 1

1. ∀ y ∈ Z such that ∃ x ∈ Z, ¬R (x + y)

2. ∃ x ∈ Z, ∀ y ∈ Z such that ¬R(x + y)

Explanation:

If we negate a quantified statement, first we negate all the quantifiers in the statement from left to right, ( keeping the same order ) then we negative the statement,

Here, the given statement,

1. ∃y ∈Z such that ∀x ∈Z, R (x + y)

By the above definition,

Negation of this statement is ∀ y ∈ Z such that ∃ x ∈ Z, ¬R (x + y),

2. Similarly,

The negation of statement ∀x ∈Z, ∃y∈Z such that R(x + y),

∃ x ∈ Z, ∀ y ∈ Z such that ¬R(x + y)