Exercises
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Exercises
Write the negations of the following statements.
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For the set D = {-3, -2, -1, 0, 1, 2, 3}, are the following statements true or false?
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Write the converse of each of the following statements and give a counterexample showing that it's false.
- For every integer x that's greater than 0, x2 is greater than 0.
- $\forall x \in\mathbb{Z}$, if x, y > 0 then $x + y > 0$.
- $\forall n \in\mathbb{N}$, if n is odd then 2n is even.
- For all acute angles, if the angles are vertical angles then they're congruent.
- For every pair of ellipses, if their centers are the same then they intersect at an even number of points.
Determine if the following statements are true or false.
- Every odd number in the set {2, 4, 6, 8, 10} is prime.
- ∀ squares, S, if S has exactly one pair of parallel sides then it's a trapezoid.
- $\forall x \in \mathbb{Z}^-$, every positive number is prime.
Rewrite the following statements in the if-then form.
- Being odd is a sufficient condition for not being even.
- Having four congruent sides is a necessary condition for being a square.
- $x \ge 0$ is a necessary condition for |x| = x.
- $x \ge 0$ is a sufficient condition for $x^2 \ge 0$.
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Write the negation of the statement, "There are real numbers that aren't positive."
Rewrite this as an existential statement first.
Write the negation of the statement, "There are real numbers that aren't positive."
First, I'll rewrite this as an existential statement since it's easier to apply the negation rule when it's in that form.
$\exists x\in \mathbb{R}$ such that x isn't positive.
From here, we can rewrite the negation as a universal statement.
$\forall x\in \mathbb{R}$, x is positive.
or
~($\exists x\in \mathbb{R}$ such that x isn't positive.) ≡ $\forall x\in \mathbb{R}$, x is positive.
Write the negation of the statement, "Every continuous function is differentiable."
Start by rewriting this as a universal (or universal conditional) statement.
Write the negation of the statement, "Every continuous function is differentiable."
If we write this as a conditional statement, it becomes
$\forall x\in \text{{continuous functions}}, x\in \text{{differentiable functions}}$
Now we can see that the negation is
$\exists x\in \text{{continuous functions}}, x\not\in \text{{differentiable functions}}$
or
~($\forall x\in \text{{continuous functions}}, x\in \text{{differentiable functions}}$) ≡ $\exists x\in \text{{continuous functions}}, x\not\in \text{{differentiable functions}}$
Write the negation of the statement, "$\forall x\in\mathbb{R}, \sqrt{x} \ge x$".
$$\exists x\in\mathbb{R}^+, \sqrt{x} \lt x$$
For the set D = {-3, -2, -1, 0, 1, 2, 3}, is $\forall x\in D, x^2 \ge 0$ true or false?
The square of every number, including the ones in A, is greater than or equal to 0 so this is a true statement.
For the set D = {-3, -2, -1, 0, 1, 2, 3}, is $\forall x, y\in D, x + y \in D$ true or false?
The technical term for this is 'closed', i.e. the set is asking if the set is closed under addition. This isn't true because, for example, when $x = 2$ and $y = 3$ $x + y = 5$ which isn't in D.
For the set D = {-3, -2, -1, 0, 1, 2, 3}, is '$\exists x\in D$, |x| is odd.' true or false?
This is true because $x = -3 \in D$ and $|-3| = 3$ which is odd.
Write the converse of 'For every integer x that's greater than 0, x2 is greater than 0.' and give a counterexample showing it's false.
Converse: For every integer x if x2 is greater than 0 then $x \gt 0$.
Consider x = -1. $(-1)^2 = 1 > 1$ but x is not greater than 0.
Write the converse of '$\forall n \in\mathbb{N}$, if n is odd then 2n is even.' and give a counterexample showing it's false.
Converse: $\forall n \in\mathbb{N}$, if 2n is even then n is odd.
Consider $n = 4$. $2n = 8$ which is even but n isn't odd.
Write the converse of 'For every pair of ellipses, if their centers are the same then they intersect at an even number of points.' and give a counterexample showing it's false.
Converse: For every pair of ellipses, if they intersect at an even number of points then their centers are the same.
Consider the ellipses $\frac{x^2}{10^2} + \frac{y^2}{9^2} = 1$ and $\frac{(x-1)^2}{1^2} + \frac{(y-1)^2}{2^2} = 1$. The second ellipse is contained in the first, i.e. they have 0 intersection points. 0 is even but their centers aren't the same.
Is "∀ squares, S, if S has exactly one pair of parallel sides then it's a trapezoid." true or false?
Squares all have two pairs of parallel sides so there are no squares that meet the criterion, "has exactly one pair of parallel sides". This makes the original statement true by default or a "vacuous truth".
Rewrite, "Being odd is a sufficient condition for not being even." in the if-then form.
Remember that the rule is "$p(x)$ is a sufficient condition for $q(x)$" is equivalent to "If $p(x)$ then $q(x)$."
Rewrite, "Being odd is a sufficient condition for not being even." in the if-then form.
Remember that the rule is "$p(x)$ is a sufficient condition for $q(x)$" is equivalent to "If $p(x)$ then $q(x)$."
That makes this statement, "If an integer, x, is odd then x is not even." or "For every integer, x, if x is odd then x is not even."
Rewrite, "$x \ge 0$ is a necessary condition for |x| = x" in the if-then form.
Remember that the rule is "$p(x)$ is a sufficient condition for $q(x)$" is equivalent to "If ~$p(x)$ then ~$q(x)$."
Rewrite, "$x \ge 0$ is a necessary condition for |x| = x" in the if-then form.
Remember that the rule is "$p(x)$ is a sufficient condition for $q(x)$" is equivalent to "If ~$p(x)$ then ~$q(x)$."
That makes this statement, "If $x \lt 0$ then $|x| \ne x$."
