91Ó°ÊÓ

Use truth tables to show that the following statements are logically equivalent. P \wedge(Q \vee R)=(P \wedge Q) \vee(P \wedge R)

Be sure to also state exactly what statements \(P\) and \(Q\) stand for. The number 8 is both even and a power of 2 .

For matrix \(A\) to be invertible, it is necessary and sufficient that \(\operatorname{det}(A) \neq 0\).

Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " A matrix is invertible provided that its determinant is not zero.

Translate each of the following sentences into symbolic logic. If \(f\) is a polynomial and its degree is greater than 2 , then \(f^{\prime}\) is not constant.

Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible. Every real number is an even integer.

Negate the following sentences. The number \(x\) is positive, but the number \(y\) is not positive.

Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " For a function to be continuous, it is sufficient that it is differentiable.

Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible. Every even integer is a real number.

Write the following as English sentences. Say whether they are true or false. $$ \forall x \in \mathbb{R}, \exists n \in \mathbb{N}, x^{n} \geq 0 $$