Miscellaneous
Summation notation
- Take the sum \(x_1+x_2+x_3+...+x_n\)
- We can express this sum using the summation symbol \(\Sigma\): \(\sum_{i=1}^n x_i=x_1+x_2+x_3+...+x_n\)
- What does the notation mean? What does the superscript mean? The subscript?
- IMPORTANT: Whenever I do not specify superscript and subscript, always assume \(i=1, n=n\)
- How do we interpret the following? \(\Sigma_{i=30}^{35}N_i\)
- Do example 1, p56
Rules of sums & Newton's binomial formula
- Additivity property: \(\Sigma(a_i+b_i)=\Sigma a_i + \Sigma b_i\)
- Homogeneity property: \(\Sigma (c\cdot a_i)=c\Sigma a_i\), where \(c\) is a constant
- From the homogeneity property, it follows that \(\Sigma c=n\cdot c\). Why?
- Do example 2 p60: Derive the fact that the sum of the difference between \(x_i\) and its arithmetic mean \(\mu_x\) is equal to zero: \(\Sigma (x_i-\mu_x)=0\)
Useful formulas
- Gauss' formula: \(\sum i= \frac{1}{2} n(n+1)\)
- Proof:
- \(x=1+2+...+(n-1)+n\)
- We can rewrite this as \(x=n+(n-1)+...+2+1\)
- From these we can write \(2x=[n+(n-1)+...+2+1]+[1+2+...+(n-1)+n]\)
- Rearranging we get \(2x=(n+1)+(n-1+2)+...+(2+n-1)+(1+n)=(n+1)+(n+1)+...+(n+1)+(n+1)\)
- Finally, we get \(2x=n\left(n+1\right)\). Solving for \(x\): \(x=\frac{1}{2}n(n+1)\). QED
- Two other useful formulas are \(\sum i^2=\frac{1}{6}n(n+1)(2n+1)\) and \(\sum i^3=[\sum i]^2\)
- Newton's binomial formula: \((a+b)^m= a^m+\left( \begin{array}{c} m \\ 1 \end{array} \right)a^{m-1}b+...+\left( \begin{array}{c} m \\ m-1 \end{array} \right)ab^{m-1}+\left( \begin{array}{c} m \\ m \end{array} \right)b^m\) where the binomial coefficients \(\left( \begin{array}{c} m \\ k \end{array} \right)=\frac{m(m-1)...(m-k+1)}{k!}\) are defined for \(m=1, 2, ...\) and \(k=0,1,2,...,m\)
- Application of Newton's formula:
- \((a+b)^3=a^3+\left( \begin{array}{c} 3 \\ 1 \end{array} \right)a^{2}b+\left( \begin{array}{c} 3 \\ 2 \end{array} \right)ab^{2}+b^3\). We can use the formula from above to write \((a+b)^3=a^3+\frac{3}{1}a^{2}b+ \frac{3\cdot 2}{1\cdot 2}ab^{2}+b^3=a^3+3a^2b+3ab^2+b^3\)
Double sums
- It's possible to calculate the sum of sums by using the following formula: \(\sum_{i=1}^m \sum_{j=1}^na_{ij}\)
- Do example 1 p65: \(\sum_{i=1}^3\sum_{j=1}^4(i+2j)=\sum_{i=1}^3[(i+2)+(i+4)+(i+6)+(i+8)]=\sum_{i=1}^3(4i+20)=(4+20)+(8+20)+(12+20)=84\)
A few aspects of logic
- Do example 1, p66
- Propositions: assertions that are either true or false. When an assertion contains one or more variables for which it can be true or false we say it is an open proposition
- The implication arrow: \(\Longrightarrow\)
- \(P\Longrightarrow Q\) you read as "p implies q" or "if p then q"
- The logical equivalence arrow: \( \Longleftrightarrow\)
- \(P \Longleftrightarrow Q\) you read as "p if and only if q"
- Do example 2 p67
- Necessary and sufficient conditions:
- If \(P \Longrightarrow Q\) we say that p is sufficient condition for q
- If \(P\Longrightarrow Q\) we say that q is necessary condition for p
Solving equations
Mathematical proofs
- Every mathematical theorem can be formulated as an implication. Indeed, all proofs of mathematical theorems rely on the establishment of implications between premises (or assumptions) and conclusions.
- Examples of proof methods:
- Direct proof: We start from the premises and we keep deriving their implications until we get to the conclusions.
- Indirect proof: We deny the conclusions and show that the premises must also be false. The indirect proof relies on the fact that \(P\Longrightarrow Q\) is equivalent to \(\backsim Q \Longrightarrow \backsim P\) (non-q implies non-p)
Essentials of set theory
- We partially covered this last week
- You should be familiar with the following notions/notations
- Set and elements: \(S= \{e_1, e_2, e_3, ..., e_n\}\)
- Two sets, A and B, are said to be equal if every element in A is also an element in B
- Do examples
Property of a set
- We can use the following notation to specify the property of a set: \(S=\{e: p\}\), where e=typical elements and p= defining properties
- Example: the budget set. \(B=\{(x, y): px+py\leq m, x\geq 0, y\geq 0\}\). Explain.
Set membership
- We covered this in class last week. Remember the meaning of the following symbols:
- \(\subset\)
- \(\subseteq\)
- \(\in\)
- \(\notin\)
Set operations
- Union: \(\cup \). All the elements that belong to at least one of the sets.
- Intersection: \(\cap\). All the elements that belong to both sets.
- Minus: \(\setminus\) .All the elements that belong to one set but not the other.
- Do example 1 p76
- Other important notions:
- Disjoint set: the empty set. For example, the intersection between two sets that do not share any members
- The universal set, \(\Omega\): The set containing all potential subsets of a family of sets.
- Complement set: \(A^c=\Omega\setminus A\). It contains all elements of the universal set not contained in A.