Translate ∀x∃y(x<y) in English. Consider domain as a real number for both the variables.
For all real numbers x, there exists a real number y such that x is less than y
Nested quantifiers are read from left to right, and their order changes the meaning. ∀x means for every x, and the following ∃y means that, for that particular x, some y exists (the choice of y may depend on x).
Step 1 — ∀x: for all real numbers x.
Step 2 — ∃y: there exists a real number y.
Step 3 — (x < y): such that x is less than y. The statement says no real number is the largest, and it is true (take y = x + 1).
The sentence beginning for every real number y swaps the roles of the variables and is the formula ∀y∃x(x < y). The sentence beginning for some real numbers x uses an existential quantifier on x, and the one saying x is less than y for each and every pair is ∀x∀y(x < y), which is false.
For all real numbers x, there exists a real number y such that x is less than y option is correct.
Similar Questions
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\(\sim(\forall x)\,F[x]\)
is
Let P(m, n) be the statement “m divides n” where the Universe of discourse for both the variables is the set of positive integers. Determine the truth values of the following propositions.
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Codes :
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\(\sim(\forall x)\,F[x]\)
is
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Codes :
