🔬 Tutorial problems alpha \alpha

🔬 Tutorial problems alpha \(\alpha\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\alpha\).1#

This is an exercise on necessary and sufficient conditions. Start with some definitions from geormetry:

  • a quadrilateral is a planar figure made up of four line segments connected to form a closed shape. Consider only simple quadrilaterals without self-intersections.

  • a quadrilateral with a pair of parallel sides is called a trapezoid

  • a quadrilateral with two pais of parallel sides is called a parallelogram

  • a parallelogram with right angles is called a rectangle

  • a rectangle with all sides of the same length is called a square

  • a parallelogram with all sides of the same length is called a rhombus

illustration

For each of the statements below determine if they are True or False. For those which are False, provide a counterexample.

  1. Sufficient condition for a quadrilateral to be a square is for it to be a rhombus.

  2. Necessary condition for a quadrilateral to be a square is for it to be a rectangle.

  3. Sufficient condition for a quadrilateral to be a rectangle is for it to be a parallelogram.

  4. Necessary condition for a quadrilateral to be a rectangle is for it to be a parallelogram.

  5. Necessary condition for a quadrilateral to be a parallelogram is for it to be a trapezoid.

  6. Sufficient condition for a quadrilateral to be a parallelogram is for it to be a square.

  7. Necessary condition for a quadrilateral to be a trapezoid is for it to be a rhombus.

  8. Necessary condition for a quadrilateral to be a rectangle is for it to be a trapezoid.

  9. Necessary and sufficient conditions for a quadrilateral to be a square are for it to simultaneously be a rectangle and a rhombus.

Review the definitions from the lecture on logic and proofs.

\(\alpha\).2#

Using direct proof, show that the difference between a cube of a natural number greater than one, \(n=2,3,\dots\), and the number itself is divisible by 6.

Factor the expression \(n^3-n\) and use the property of the consecutive natural numbers.

\(\alpha\).3#

Prove by contradiction that \(\sqrt{3}\) is an irrational number (so that is it can not be represented by a ratio of two whole numbers).

Assume the opposite, namely that \(\sqrt{3} = \frac{p}{q}\) where without loss of generality whole numbers \(p\) and \(q\) do not have common factors.

\(\alpha\).4#

Using mathematical induction prove that \(n < 2^n\) holds for all natural numbers \(n=1,2,\dots\).

Verify the statement holds for \(n=1\), and assuming \(n<2^n\) prove that \(n+1 < 2^{n+1}\).

\(\alpha\).5#

Let \(A\),\(B\) and \(C\) be any three sets. Show that \(A \cap (B \cup C) = (A\cap B) \cup (A \cap C)\).

One way to show that \(E=F\) is show that a arbitrary element of \(E\) must also be in \(F\) and vice versa.

\(\alpha\).6#

Find the composition \(g \circ f\) of two functions \(f\) and \(g\), if it exists:

  1. \(f \colon \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=\sin(x)\) and \(g \colon \mathbb{R} \rightarrow \mathbb{R}\) defined by \(g(x)= \frac{x}{1+x^2}\)

  2. \(f \colon \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)= 1-x^4\) and \(g \colon (1,\infty) \rightarrow \mathbb{R}\) defined by \(g(x)= \log(x-1)\)

  3. \(f \colon \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=\cos(x)\) and \(g \colon \mathbb{R}\setminus\{1\} \rightarrow \mathbb{R}\) defined by \(g(x)= \frac{x}{1-x}\)

Is there a composition in each case?

\(\alpha\).7#

For each of the following functions, what is its domain and range? Is it one-to-one? Is it onto? Is it bijective? If it is bijective, write the expression for the inverse function.

  1. \(f(x) = 3x-7\)

  2. \(f(x) = e^x\)

  3. \(f(x) = \ln(x)\)

  4. \(f(x) = \sqrt{x-1}\)

  5. \(f(x) = x^2-1\)

Draw a graph for each function.