Elementary logic

Write the negation of

$$\forall x\in \mathbb{R}:x^2\ge 1$$

and determine which of the proposition and its negation is true.

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Write the negation of

$$\exists n\in \mathbb{N}:n^2=10$$

and determine which of the proposition and its negation is true.

Solution available to those who have unlocked the subject.

Write the negation of

$$\forall x\in \mathbb{R}:x\ge 0\wedge x^2\ge 0$$

and determine which of the proposition and its negation is true.

Solution available to those who have unlocked the subject.

Write the negation of

$$\exists y\in \mathbb{R}\forall x\in \mathbb{R}:x^2\ge y$$

and determine which of the proposition and its negation is true.

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Determine whether, for natural $n$, the condition “being a multiple of $9$” is necessary, sufficient, or necessary and sufficient for “being a multiple of $3$”.

Solution available to those who have unlocked the subject.

Determine whether, for real $x$, the condition "$x^2>0$" is necessary, sufficient, or necessary and sufficient for "$x>0$".

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Determine whether, for real $x$, the condition "$2x-4=0$" is necessary, sufficient, or necessary and sufficient for "$x=2$".

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Write the contrapositive and the converse of the implication “if $n$ is a multiple of $8$, then $n$ is a multiple of $4$” ($n$ natural) and determine the truth value of each.

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Write the negation of

$$\forall x\in \mathbb{R}:x>0⟹x+1>2$$

and determine which of the proposition and its negation is true.

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Given that $\sqrt{2}$ is irrational, prove by contradiction that $1+\sqrt{2}$ is irrational.

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