Quantifiers Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

Write in words: (a)

\forall x \in \mathbb{Z},\; x + 0 = x

, (b)

\exists x \in \mathbb{N},\; x < 5

1
(a) 'For every integer $x$ , $x + 0 = x$ .' (The additive identity law — True.)
2
(b) 'There exists a natural number $x$ such that $x < 5$ .' (For example $x = 1$ — True.)

Answer

(a)\;\text{Every integer plus zero equals itself.}\quad (b)\;\text{Some natural number is less than 5.}

Translating symbolic quantifier statements to plain English clarifies their meaning. Identifying one witness (like

x=1

) is sufficient to confirm an existential claim.

About Quantifiers

Symbols specifying the scope of a predicate: $\forall$ (for all, universal) and $\exists$ (there exists, existential).

Translate into symbols and determine the truth value: (a) 'Every natural number is positive.', (b) '

Negate the statement [formula] and determine the truth value of both the original and its negation.

Determine the truth value of each and write its negation: (a) [formula], (b) [formula].