집합과 명제 전체 문제

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

$\left(a+b \right)^2=a^2+b^2$, for all real numbers $a$ and $b$.

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

There are integers $x$ and $y$ such that $2x+5y=41$.

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

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

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

If $x$ and $y$ are odd integers, then $x \cdot y$ is an odd integer.

Determine whether each of the following conditional statements is true or false.

If $10 \lt 7$, then $3 = 4$.

Determine whether each of the following conditional statements is true or false.

If $7 \lt 10$, then $3 = 4$.

Determine whether each of the following conditional statements is true or false.

If $10 \lt 7$, then $3 + 5 = 8$.

Determine whether each of the following conditional statements is true or false.

If $7 \lt 10$, then $3 + 5 =8$.

Suppose that Daisy says, “If it does not rain, then I will play golf.” Later in the day you come to know that it did rain but Daisy still played golf. Was Daisy's statement true or false? Support your conclusion.

Suppose that $P$ and $Q$ are statements for which $Q$ is false and $\neg P \to Q$ is true (and it is not known if $R$ is true or false). What conclusion (if any) can be made about the truth value of each of the following statements?

$\neg Q \to P$

Suppose that $P$ and $Q$ are statements for which $Q$ is false and $\neg P \to Q$ is true (and it is not known if $R$ is true or false). What conclusion (if any) can be made about the truth value of each of the following statements?

$P$

Suppose that $P$ and $Q$ are statements for which $Q$ is false and $\neg P \to Q$ is true (and it is not known if $R$ is true or false). What conclusion (if any) can be made about the truth value of each of the following statements?

$P \wedge R$

Suppose that $P$ and $Q$ are statements for which $Q$ is false and $\neg P \to Q$ is true (and it is not known if $R$ is true or false). What conclusion (if any) can be made about the truth value of each of the following statements?

$R \to \neg P$

Assume that the universal set is $\mathbb{Z}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{Z}} \right)\left( {t \cdot x = 20} \right)$$

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If 5 is substituted for $x$, is the resulting sentence a statement? If it is a statement, is the statement true or false?

Assume that the universal set is $\mathbb{Z}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{Z}} \right)\left( {t \cdot x = 20} \right)$$

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If 8 is substituted for $x$, is the resulting sentence a statement? If it is a statement, is the statement true or false?

Assume that the universal set is $\mathbb{Z}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{Z}} \right)\left( {t \cdot x = 20} \right)$$

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If $-2$ is substituted for $x$, is the resulting sentence a statement? If it is a statement, is the statement true or false?

Assume that the universal set is $\mathbb{R}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{R}} \right)\left( {t \cdot x = 20} \right)$$

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If 5 is substituted for $x$, is the resulting sentence a statement? If it is a statement, is the statement true or false?

Assume that the universal set is $\mathbb{R}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{R}} \right)\left( {t \cdot x = 20} \right)$$

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If $\pi$ is substituted for $x$, is the resulting sentence a statement? If it is a statement, is the statement true or false?

Assume that the universal set is $\mathbb{R}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{R}} \right)\left( {t \cdot x = 20} \right)$$

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If $0$ is substituted for $x$, is the resulting sentence a statement? If it is a statement, is the statement true or false?

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For all integers $a$, $b$, and $c$ with $a \ne 0$, if $a \mid b$, then $a \mid \left(bc \right)$.