Mathematical Reasoning

$\sim p \vee q$

$\sim p \wedge q$

$p \wedge \sim q$

$\sim (p \wedge q)$

$x$ is a fraction

$x$ is an integer

$x$ is a natural number

None of these

True

False

Neither

None of these

Existential statement

Universal statement

Implication

Negation

$\sim q$

$p$

$q$

$\sim p$

$\\sim p \\vee q$

$\\sim p \\wedge q$

$p \\vee \\sim q$

$\\sim p \\vee \\sim q$

If $x + c = y + c$, then $x = y$.

If $x \\neq y$, then $x + c \\neq y + c$.

If $x + c \\neq y + c$, then $x \\neq y$.

If $x = y$, then $x + c \\neq y + c$.

$[(p \rightarrow q) \wedge (q \rightarrow r)] \rightarrow (p \rightarrow r)$

$[(p \vee q) \wedge \sim p] \rightarrow q$

$p \rightarrow (p \vee q)$

$(p \wedge q) \rightarrow p$

If $f$ is not differentiable, then $f$ is not continuous.

If $f$ is not continuous, then $f$ is not differentiable.

If $f$ is continuous, then $f$ is differentiable.

If $f$ is differentiable, then $f$ is not continuous.

$(p \wedge \sim q) \vee (q \wedge \sim p)$

$(p \vee \sim q) \wedge (q \vee \sim p)$

$\sim p \leftrightarrow \sim q$

$\sim p \wedge \sim q$

The converse

The inverse

The original implication

The negation

If $x + z = y + z$, then $x = y$

If $x \neq y$, then $x + z \neq y + z$

If $x + z \neq y + z$, then $x \neq y$

If $x = y$, then $x + z \neq y + z$

If the square is positive, the number is positive

If the square is not positive, the number is not positive

If a number is not positive, its square is not positive

If the number is positive, its square is not positive

If the square of the hypotenuse is the sum of squares of other sides, the triangle is right-angled

If the square of the hypotenuse is not the sum of squares of other sides, the triangle is not right-angled

If a triangle is not right-angled, then the square of hypotenuse is not the sum of squares of other sides

If the square of the hypotenuse is not the sum of squares of other sides, the triangle is right-angled

If not all sides of a quadrilateral are equal, then it is not a square

If all sides of a quadrilateral are equal, then it is a square

If a quadrilateral is not a square, then its sides are not equal

If some sides of a quadrilateral are equal, then it is a square

$p \land q$

$p \land \neg q$

$\neg p \lor \neg q$

$q \Rightarrow \neg p$

If $x^2 = 4$, then $x = 2$

If $x \neq 2$, then $x^2 \neq 4$

If $x^2 \neq 4$, then $x \neq 2$

If $x = -2$, then $x^2 = 4$

If $a-c \le b-c$, then $a \le b$.

If $a \le b$, then $a-c \le b-c$.

If $a-c > b-c$, then $a > b$.

If $a < b$, then $a-c < b-c$.

If a polygon is not a square, then it is not a rhombus.

If a polygon is a rhombus, then it is a square.

If a polygon is not a rhombus, then it is not a square.

If it is not a square, it might be a rhombus.

$\neg r \Rightarrow (p \land \neg q)$

$\neg r \Rightarrow (\neg p \land q)$

$\neg r \Rightarrow (\neg p \lor q)$

$r \Rightarrow (p \lor \neg q)$

No birds can fly

Some birds cannot fly

All birds cannot fly

Most birds can fly

Fallacy

Tautology

Contradiction

Theorem

True

False

Both

None of these

If you do not study hard, you will not pass

If you pass, you studied hard

If you do not pass, you did not study hard

If you study hard, you will not pass

$x^2$ is even and $x$ is even

$x^2$ is odd and $x$ is odd

$x^2$ is even and $x$ is odd

$x^2$ is odd and $x$ is even

$x$ is not rational and $y$ is not an integer.

$x$ is not rational or $y$ is not an integer.

$x$ is rational or $y$ is not an integer.

If $x$ is not rational, then $y$ is not an integer.

Logically True

Logically False

A contradiction

An inverse

$p \iff q$

$\sim p \iff q$

$p \implies q$

$q \implies p$

$p \iff q$

$\sim p \vee \sim q$

$p \implies \sim q$

$\sim p \wedge q$

There exists a prime $n$ such that $\\sqrt{n}$ is rational

For all primes $n$, $\\sqrt{n}$ is rational

For all primes $n$, $\\sqrt{n}$ is irrational

There exists a prime $n$ such that $\\sqrt{n}$ is irrational