Lesson test: Statements and connectives

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Statements and connectives

1.[1p]
Which of these is a statement?

$91$ is prime

Correct

$x^{2} = 4$

The answer is: $91$ is prime

Is $91$ prime?

The answer is: $91$ is prime

Let $n = 91$ .

The answer is: $91$ is prime

2.[3p]
Match each compound statement to the only circumstances in which it has the stated truth value.

$\neg P$ is true
$P \land Q$ is true
$P \lor Q$ is false
$P \Rightarrow Q$ is false
$P ⟺ Q$ is true

only when $P$ and $Q$ are both true
exactly when $P$ is false
only when $P$ is true and $Q$ is false
only when $P$ and $Q$ are both false
exactly when $P$ and $Q$ have the same truth value

Show the answer

$\neg P$ is true: exactly when $P$ is false $P \land Q$ is true: only when $P$ and $Q$ are both true $P \lor Q$ is false: only when $P$ and $Q$ are both false $P \Rightarrow Q$ is false: only when $P$ is true and $Q$ is false $P ⟺ Q$ is true: exactly when $P$ and $Q$ have the same truth value

3.[1p]
The statement "if $2 + 2 = 5$ , then $7$ is even" is true.

True

Correct

False

The answer is: True

4.[2p]
What is the negation of " $x < 0$ or $x > 1$ "?

$x \geq 0$ and $x \leq 1$

Correct

$x \geq 0$ or $x \leq 1$

The answer is: $x \geq 0$ and $x \leq 1$

$x > 0$ and $x < 1$

The answer is: $x \geq 0$ and $x \leq 1$

$x \geq 0$ and $x > 1$

The answer is: $x \geq 0$ and $x \leq 1$

5.[2p]
What is the negation of "if $n^{2}$ is even, then $n$ is even"?

$n^{2}$ is even and $n$ is odd

Correct

If $n^{2}$ is even, then $n$ is odd

The answer is: $n^{2}$ is even and $n$ is odd

If $n^{2}$ is odd, then $n$ is odd

The answer is: $n^{2}$ is even and $n$ is odd

$n^{2}$ is odd or $n$ is even

The answer is: $n^{2}$ is even and $n$ is odd

6.[3p]
Which of these are equivalent to $P \Rightarrow Q$ ?

Select all that apply

$\neg Q \Rightarrow \neg P$

Correct

$P$ only if $Q$

Correct

$Q$ is a necessary condition for $P$

Correct

$Q \Rightarrow P$

The answer is: $\neg Q \Rightarrow \neg P$ , $P$ only if $Q$ , $Q$ is a necessary condition for $P$

$\neg P \Rightarrow \neg Q$

The answer is: $\neg Q \Rightarrow \neg P$ , $P$ only if $Q$ , $Q$ is a necessary condition for $P$

$P$ if $Q$

The answer is: $\neg Q \Rightarrow \neg P$ , $P$ only if $Q$ , $Q$ is a necessary condition for $P$

7.[2p]
Take the statement "if $n$ is divisible by $10$ , then $n$ is divisible by $5$ ". Match each name to the conditional it names.

Converse
Inverse
Contrapositive

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

Show the answer

Converse: if $n$ is divisible by $5$ , then $n$ is divisible by $10$ Inverse: if $n$ is not divisible by $10$ , then $n$ is not divisible by $5$ Contrapositive: if $n$ is not divisible by $5$ , then $n$ is not divisible by $10$

8.[1p]
Being divisible by $3$ is a sufficient condition for an integer to be divisible by $6$ .

True

The answer is: False

False

Correct

9.[2p]
Put these steps of the argument that a conditional with a false hypothesis must count as true in order.

"For every integer $n$ , if $n$ is divisible by $4$ , then $n$ is even" is a true theorem.
A claim about every integer is true only if its instance for each integer is true.
So a conditional with a false hypothesis must count as true, whatever its conclusion.
So the instance at $n = 6$ , with a false hypothesis and a true conclusion, and the instance at $n = 7$ , with both parts false, must both be true.

Show the answer

a, b, c, d

10.[1p]
The statement $(P \land \neg P) \Rightarrow Q$ is a tautology, whatever statement $Q$ is.

True

Correct

False

The answer is: True

11.[2p]
What does " $P$ unless $Q$ " mean?

$\neg Q \Rightarrow P$

Correct

$Q \Rightarrow \neg P$

The answer is: $\neg Q \Rightarrow P$

$P ⟺ \neg Q$

The answer is: $\neg Q \Rightarrow P$

$P \land \neg Q$

The answer is: $\neg Q \Rightarrow P$

12.[2p]
Which of these argument patterns is invalid?

From $P \Rightarrow Q$ and $Q$ , conclude $P$

Correct

From $P \Rightarrow Q$ and $P$ , conclude $Q$

The answer is: From $P \Rightarrow Q$ and $Q$ , conclude $P$

From $P \Rightarrow Q$ and $\neg Q$ , conclude $\neg P$

The answer is: From $P \Rightarrow Q$ and $Q$ , conclude $P$

From $P \lor Q$ and $\neg P$ , conclude $Q$

The answer is: From $P \Rightarrow Q$ and $Q$ , conclude $P$

13.[3p]
Which of these sets of connectives are functionally complete?

Select all that apply

${\neg, \land}$

Correct

${\neg, \Rightarrow}$

Correct

${↑}$ (NAND)

Correct

${\land, \lor}$

The answer is: ${\neg, \land}$ , ${\neg, \Rightarrow}$ , ${↑}$ (NAND)

${\Rightarrow}$

The answer is: ${\neg, \land}$ , ${\neg, \Rightarrow}$ , ${↑}$ (NAND)

${\neg, ⟺}$

The answer is: ${\neg, \land}$ , ${\neg, \Rightarrow}$ , ${↑}$ (NAND)

14.[1p]
How many different truth functions of two variables are there, that is, how many different four-row truth tables?

CorrectNot quite: 16

1.[1p] Which of these is a statement?

2.[3p] Match each compound statement to the only circumstances in which it has the stated truth value.

3.[1p] The statement "if 2+2=5, then 7 is even" is true.

4.[2p] What is the negation of "x<0 or x>1"?

5.[2p] What is the negation of "if n2 is even, then n is even"?

6.[3p] Which of these are equivalent to P⇒Q?

7.[2p] Take the statement "if n is divisible by 10, then n is divisible by 5". Match each name to the conditional it names.

8.[1p] Being divisible by 3 is a sufficient condition for an integer to be divisible by 6.

9.[2p] Put these steps of the argument that a conditional with a false hypothesis must count as true in order.

10.[1p] The statement (P∧¬P)⇒Q is a tautology, whatever statement Q is.

11.[2p] What does "P unless Q" mean?

12.[2p] Which of these argument patterns is invalid?

13.[3p] Which of these sets of connectives are functionally complete?

14.[1p] How many different truth functions of two variables are there, that is, how many different four-row truth tables?

1.[1p]
Which of these is a statement?

2.[3p]
Match each compound statement to the only circumstances in which it has the stated truth value.

3.[1p]
The statement "if $2 + 2 = 5$ , then $7$ is even" is true.

4.[2p]
What is the negation of " $x < 0$ or $x > 1$ "?

5.[2p]
What is the negation of "if $n^{2}$ is even, then $n$ is even"?

6.[3p]
Which of these are equivalent to $P \Rightarrow Q$ ?

7.[2p]
Take the statement "if $n$ is divisible by $10$ , then $n$ is divisible by $5$ ". Match each name to the conditional it names.

8.[1p]
Being divisible by $3$ is a sufficient condition for an integer to be divisible by $6$ .

9.[2p]
Put these steps of the argument that a conditional with a false hypothesis must count as true in order.

10.[1p]
The statement $(P \land \neg P) \Rightarrow Q$ is a tautology, whatever statement $Q$ is.

11.[2p]
What does " $P$ unless $Q$ " mean?

12.[2p]
Which of these argument patterns is invalid?

13.[3p]
Which of these sets of connectives are functionally complete?

14.[1p]
How many different truth functions of two variables are there, that is, how many different four-row truth tables?