Examples

Prove using rules of inference that:
$(𝑃 \Rightarrow \neg 𝑄) \land (𝑄 \Rightarrow \neg 𝑃) \equiv \neg 𝑃 \lor \neg 𝑄 .$

Solution.

\begin{matrix} (𝑃 \Rightarrow \neg 𝑄) \land (𝑄 \Rightarrow \neg 𝑃) \\ \equiv (\neg 𝑃 \lor \neg 𝑄) \land (\neg 𝑄 \lor \neg 𝑃) & by Material Implication \\ \equiv (\neg 𝑃 \lor \neg 𝑄) \land (\neg 𝑃 \lor \neg 𝑄) & by Commutativity of \lor \\ \equiv \neg 𝑃 \lor \neg 𝑄 . & by Idempotent Law & ∎ \end{matrix}

Prove that the following argument is invalid:

If this number is larger than $9$ , then its square is larger than $81$ .
The number is not larger than $9$ .
Therefore, its square is not larger than $81$ .

Solution.

Let $𝑝$ be the proposition “This number is larger than $9$ ” and let $𝑞$ be the proposition “Its square is larger than $81$ ”. The argument can be rewritten as follows:

If $𝑝$ , then $𝑞$ .
Not $𝑝$ .
Therefore, not $𝑞$ .

The statement $𝑝 \Rightarrow 𝑞$ is not equivalent to $\neg 𝑝 \Rightarrow \neg 𝑞$ . The argument is invalid by inverse error.

Proof.

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Proof: Solution of .

1. Fields

Definition 1: Well Ordering Principle

Let $𝑥$ be a nonempty set of nonnegative integers. Then $𝑥$ has a least element.

Definition 2: Field

A field is a set $𝐹$ together with two binary operations addition $+$ and multiplication $\cdot$ such that the following properties hold:

Closure under addition. $\forall 𝑎, 𝑏 \in 𝐹, 𝑎 + 𝑏 \in 𝐹 .$
Closure under multiplication. $\forall 𝑎, 𝑏 \in 𝐹, 𝑎 \cdot 𝑏 \in 𝐹 .$
Commutativity of addition. $\forall 𝑎, 𝑏 \in 𝐹, 𝑎 + 𝑏 = 𝑏 + 𝑎 .$
Commutativity of multiplication. $\forall 𝑎, 𝑏 \in 𝐹, 𝑎 \cdot 𝑏 = 𝑏 \cdot 𝑎 .$
Associativity of addition. $\forall 𝑎, 𝑏, 𝑐 \in 𝐹, (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) .$
Associativity of multiplication. $\forall 𝑎, 𝑏, 𝑐 \in 𝐹, (𝑎 \cdot 𝑏) \cdot 𝑐 = 𝑎 \cdot (𝑏 \cdot 𝑐) .$
Additive identity. $\exists 0 \in 𝐹 s.t. \forall 𝑎 \in 𝐹, 𝑎 + 0 = 𝑎 .$
Multiplicative identity. $\exists 1 \in 𝐹 s.t. \forall 𝑎 \in 𝐹, 𝑎 \cdot 1 = 𝑎 .$
Additive inverses. $\forall 𝑎 \in 𝐹, \exists (- 𝑎) \in 𝐹 s.t. 𝑎 + (- 𝑎) = 0 .$
Multiplicative inverses. $\forall 𝑎 \in 𝐹, 𝑎 \neq 0 \Rightarrow \exists 𝑎^{- 1} \in 𝐹 s.t. 𝑎 \cdot 𝑎^{- 1} = 1 .$
Distributivity over addition. $\forall 𝑎, 𝑏, 𝑐 \in 𝐹, 𝑎 \cdot (𝑏 + 𝑐) = (𝑎 \cdot 𝑏) + (𝑎 \cdot 𝑐) .$
Distributivity over multiplication. $\forall 𝑎, 𝑏, 𝑐 \in 𝐹, (𝑎 + 𝑏) \cdot 𝑐 = (𝑎 \cdot 𝑐) + (𝑏 \cdot 𝑐) .$

Together, these are called the field axioms.

The elements of $𝐹$ are called $𝑭$ -scalars. The additive identity is denoted $0_{𝐹}$ and the multiplicative identity is denoted $1_{𝐹}$ .

Definition 3: Field Subtraction and Field Division

Let $𝐹$ be a field. The binary operation of subtraction $-$ on $𝐹$ is defined as follows:

\forall 𝑎, 𝑏 \in 𝐹, 𝑎 - 𝑏 = 𝑎 + (- 𝑏) .

The binary operation of division $/$ on $𝐹$ is defined as follows:

\forall 𝑎, 𝑏 \in 𝐹, \frac{𝑎}{𝑏} = 𝑎 \cdot 𝑏^{- 1} .

Definition 4: Subfield

Let $𝐹$ be a field, and let $𝐻$ be a subset of $𝐹$ . Then $𝐻$ is a subfield of $𝐹$ iff $𝐻$ is itself a field under the same operations of addition and multiplication as $𝐹$ .

Theorem 1: Subfield Theorem

Let $𝐹$ be a field, and let $𝐻$ be a subset of $𝐹$ . Then $𝐻$ is a subfield of $𝐹$ if and only if:

Non-emptiness. $𝐻$ is non-empty, i.e. $𝐻 \neq \emptyset$ .
Closure under subtraction. $\forall 𝑎, 𝑏 \in 𝐻, 𝑎 - 𝑏 \in 𝐻$ .
Closure under division. $\forall 𝑎, 𝑏 \in 𝐻, 𝑏 \neq 0 \Rightarrow 𝑎 / 𝑏 \in 𝐻 .$

Proof: Subfield Theorem.

Suppose $𝐻$ is a subset of $𝐹$ .

( $\Rightarrow$ ) Assume $𝐻$ is a subfield of $𝐹$ . Then $𝐻$ is a field under the same operations as $𝐹$ , so $𝐻$ must satisfy all the field axioms. In particular, $𝐻$ must be non-empty since it contains at least the additive identity. $𝐻$ must also be closed under subtraction and division since these operations are defined in terms of addition and multiplication, which are closed in $𝐻$ .

( $\Leftarrow$ ) Assume that $𝐻$ is non-empty, closed under subtraction, and closed under division. We will show that $𝐻$ satisfies all the field axioms under the same operations as $𝐹$ . Since $𝐻$ is a subset of $𝐹$ , the operations of addition and multiplication on $𝐻$ are inherited from $𝐹$ .

Proposition 1

The set of all real numbers $ℝ$ is a field.

Theorem 2: Cramer’s Rule

Let $𝐴$ be an invertible $𝑛 \times 𝑛$ matrix, and let $𝑏$ be a column vector of size $𝑛$ . Then the unique solution to the system of equations $𝐴 𝑥 = 𝑏$ is given by:

𝑥_{𝑖} = \frac{\det (𝐴_{𝑖})}{\det (𝐴)}

where $𝐴_{𝑖}$ is the matrix obtained by replacing the $𝑖$ -th column of $𝐴$ with the column vector $𝑏$ .

typst test 1

Examples

1. Fields

Definition 1: Well Ordering Principle

Definition 2: Field

Definition 3: Field Subtraction and Field Division

Definition 4: Subfield

Theorem 1: Subfield Theorem

Proposition 1

Theorem 2: Cramer’s Rule

Proposition 2

Lemma 1

Corollary

Remark

Important

Warning

Caution

Note