Mathematical Reasoning

False

Neither true nor false

True

Both true and false

p: It is raining; q: It is cold

p: It is raining; q: It is not cold

p: It is not raining; q: It is cold

p: It is raining or cold; q: none

True

False

Sometimes

Undefined

All real numbers are complex numbers.

Some real numbers are complex numbers.

No real number is a complex number.

All real numbers are not complex numbers.

There exists a natural number $n$ such that if $n$ is prime and $n > 2$, then $n$ is even.

For every natural number $n$, $n$ is prime and $n > 2$ and $n$ is even.

There exists a natural number $n$ such that $n$ is prime, $n > 2$ and $n$ is even.

For every natural number $n$, if $n$ is not prime or $n \leq 2$, then $n$ is even.

Universal quantifier

Existential quantifier

Equality quantifier

Numerical quantifier

Every student in the class has failed.

No student in the class has passed.

At least one student in the class has failed.

Some students in the class have passed.

$(r \rightarrow h) \wedge (\neg r \rightarrow p)$

$(r \rightarrow h) \vee (\neg r \rightarrow p)$

$r \rightarrow (h \wedge p)$

$(r \wedge h) \vee (\neg r \wedge p)$

2

3

4

5

$(p \wedge (p \rightarrow q)) \rightarrow q$

$(p \vee (p \rightarrow q)) \rightarrow q$

$(p \wedge (q \rightarrow p)) \rightarrow q$

$(q \wedge (p \rightarrow q)) \rightarrow p$

Prime

Composite

And

Is

Some rectangles are not quadrilaterals

No rectangle is a quadrilateral

All quadrilaterals are rectangles

It is false that all rectangles are not quadrilaterals

p: Input is high; q: Output is low

p: Input is low; q: Output is high

p: Input; q: Output

p: High; q: Low

There is a real number $x$ for which $x^2 = -1$

For all real numbers $x$, $x^2 = -1$

There is a complex number $x$ such that $x^2 = -1$

It is false that $x^2 = -1$ for some $x$

$p$ is true or $q$ is true

$p$ is true and $q$ is true

$p$ is true or $q$ is false

$p$ is false and $q$ is true

$p: \\sqrt{7}$ is irrational, $q: \\sqrt{8}$ is rational

$p: 7$ is prime, $q: 8$ is composite

$p: \\sqrt{7} \\notin \\mathbb{Q}$, $q: \\sqrt{8} \\in \\mathbb{Q}$

Both A and C

For all $x \\in \\mathbb{R}, x+3=10$

For all $x \\in \\mathbb{R}, x+3 \\neq 10$

There exists $x \\in \\mathbb{R}$ such that $x+3 \\neq 10$

There is no $x \\in \\mathbb{R}$ such that $x+3=10$

Every triangle is either not right-angled or not isosceles.

No triangle is right-angled and no triangle is isosceles.

Every triangle is not right-angled and not isosceles.

There exists a triangle which is neither right-angled nor isosceles.

$f$ is even and there exists some $x$ such that $f(x) \neq f(-x)$.

If $f$ is not even, then $f(x) \neq f(-x)$ for some $x$.

$f(x) = f(-x)$ for all $x$ and $f$ is not even.

$f$ is not even and $f(x) \neq f(-x)$ for all $x$.

There exist $x, y \in \mathbb{R}$ such that $x+y \neq y+x$.

For all $x, y \in \mathbb{R}$, $x+y \neq y+x$.

There exists $x \in \mathbb{R}$ such that for all $y \in \mathbb{R}, x+y \neq y+x$.

There exists $y \in \mathbb{R}$ such that for all $x \in \mathbb{R}, x+y \neq y+x$.

True

False

Both

None

There exists

For all

Such that

If and only if

True

False

Invalid

Infinite

$x$ is not an even number or $y$ is not an odd number

$x$ is an odd number and $y$ is an even number

$x$ is not an even number and $y$ is not an odd number

$x$ is an even number and $y$ is an odd number

p: $\\sqrt{2}$ is a real number, q: $\\pi$ is a rational number

p: $\\sqrt{2}$ is a real number, q: $\\pi$ is not a rational number

p: $\\sqrt{2}$ is an irrational number, q: $\\pi$ is an irrational number

p: $\\sqrt{2}$ is a complex number, q: $\\pi$ is a real number

There exists a real number $x$ such that $x \le 0$ and $x \ge 0$ and $x \neq 0$.

There exists a real number $x$ such that $x \le 0$ or $x \ge 0$ or $x \neq 0$.

For some real number $x$, $x$ is not greater than $0$, not less than $0$, and not equal to $0$.

No real number satisfies the condition.

True

False

Cannot be said

Both

For every natural number $n$, $n$ is not a perfect square or $n$ is not a perfect cube.

For every natural number $n$, $n$ is not a perfect square and $n$ is not a perfect cube.

There exists a natural number $n$ such that $n$ is not a perfect square or $n$ is not a perfect cube.

No natural number is both a perfect square and a perfect cube.

There exists $x$ such that $P(x)$ and not $Q(x)$ and not $R(x)$.

There exists $x$ such that $P(x)$ and (not $Q(x)$ or not $R(x)$).

For all $x$, not $P(x)$ and ($Q(x)$ or $R(x)$).

There exists $x$ such that not $P(x)$ or ($Q(x)$ and $R(x)$).

There exists a composite number that has no prime factors.

Every composite number has no prime factors.

Some prime numbers have no composite factors.

No composite number has any prime factors.

If I am happy, then you are happy.

If you are not happy, then I am not happy.

If I am not happy, then you are not happy.

If I am happy, then you are not happy.

Necessary

Sufficient

Necessary and Sufficient

Contradictory

If $n$ is not divisible by 10, then $n$ is not divisible by 5.

If $n$ is divisible by 5, then $n$ is divisible by 10.

If $n$ is not divisible by 5, then $n$ is not divisible by 10.

If $n$ is divisible by 10, then $n$ is not divisible by 5.

If $x^2$ is an odd integer, then $x$ is an odd integer.

If $x$ is not an odd integer, then $x^2$ is not an odd integer.

If $x^2$ is not an odd integer, then $x$ is not an odd integer.

If $x^2$ is an even integer, then $x$ is an even integer.

$p$ is both necessary and sufficient for $q$

$p$ is necessary but not sufficient for $q$

$p$ is sufficient but not necessary for $q$

$p$ is neither necessary nor sufficient for $q$

If $a-b \leq 0$, then $a \leq b$

If $a \leq b$, then $a-b \leq 0$

If $a-b > 0$, then $a > b$

If $a-b \leq 0$, then $a > b$

$p$ is true and $q$ is false

$p$ is false and $q$ is true

$p$ is true and $q$ is true

$p$ is false and $q$ is false

$p \iff q$

$p \implies q$

$p \lor q$

$\neg p \land \neg q$

If $x$ is not divisible by 5, then it is not divisible by 10.

If $x$ is divisible by 5, then it is divisible by 10.

If $x$ is not divisible by 10, then it is not divisible by 5.

If $x$ is not divisible by 5, then it is divisible by 10.

Contradiction

Tautology

Contingency

Equivalent to $p \land q$

3 and 5

5 and 7

2 and 3

7 and 11

No students are lazy.

All students are lazy.

Some students are not lazy.

Most students are lazy.

If $-x > -y$, then $x < y$.

If $-x \le -y$, then $x \ge y$.

If $x \ge y$, then $-x \le -y$.

If $-x < -y$, then $x > y$.

Existential quantifier

Universal quantifier

Logical quantifier

None of these

All real numbers are rational.

No real number is rational.

Some real numbers are not rational.

All rational numbers are real.

Only if $n$ is even, then $n^2$ is even.

Only if $n^2$ is even, then $n$ is even.

Both: If $n$ is even, then $n^2$ is even, and if $n^2$ is even, then $n$ is even.

That 4 is an even number.

$\\sim p \\implies \\sim q$

$q \\implies p$

$\\sim q \\implies \\sim p$

$\\sim q \\implies p$

Since $(x+1/2)^2 \\ge 0$, the sum is at least $3/4$, which is $> 0$.

The statement is only true for $x > 0$.

The discriminant is positive, so the statement is true.

The statement is a counter-example to itself.

To prove that a 'For all' statement is false.

To prove that an 'Exists' statement is true.

To prove the contrapositive of a statement.

To show that a statement is true in at least one case.

If $n$ is not divisible by 3, then $n^2$ is not divisible by 3.

If $n^2$ is not divisible by 3, then $n$ is not divisible by 3.

If $n$ is divisible by 3, then $n^2$ is divisible by 3.

If $n$ is not divisible by 3, then $n^2$ is divisible by 3.