Relational Algebra Operations

Relational Algebra Overview

• Relational algebra expressions take relations as input and return a relation as output.
• Relational algebra expressions can be combined into a relational algebra query.

Selection Operation

• Selects tuples from a relation based on a predicate.
• Notation: $\sigma_{predicate}(relation)$
• Predicates can involve $=, \neq, <, >, \geq$ and connectives like $\land$ (and), $\lor$ (or), $\lnot$ (not).
• Example: $\sigma_{branch_name = "Perryridge"}(account)$ finds accounts in the Perryridge branch.

Projection Operation

• Selects specified attributes from a relation.
• Notation: $\Pi_{attribute1, attribute2,...}(relation)$
• Example: $\Pi_{branch_name, amount}(account)$ eliminates the account_number attribute.

Union Operation

• Notation: $r \cup s$
• Conditions for validity: same arity, compatible attribute domains.
• Example: $\Pi_{customer_name}(borrower) \cup \Pi_{customer_name}(depositor)$ finds customers with an account or loan.

Set-Intersection Operation

• Notation: $r \cap s$
• Conditions for validity: same arity, compatible attribute domains.
• Example: $\Pi_{customer_name}(borrower) \cap \Pi_{customer_name}(depositor)$ finds customers with both an account and loan.

Set-Difference Operation

• Notation: $r - s$
• Conditions for validity: same arity, compatible attribute domains.
• Example: $\Pi_{customer_name}(depositor) - \Pi_{customer_name}(borrower)$ finds customers with an account but not a loan.

Cartesian-Product Operation

• Notation: $r \times s$
• Requires disjoint attributes between r and s; renaming is needed if not disjoint.

Rename Operation

• Renames a relational algebra expression.
• Notation: $\rho_{x(A1, A2,..., An)}(E)$
• Renames expression $E$ to $x$ with attributes $A_1, A_2,..., A_n$.
• Example: $\rho_{cust_borrower}(\Pi_{customer_name}(borrower))$.

Theta-Join

• $r \Join_{\theta} s = \sigma_{\theta}(r \times s)$
• Involves a predicate $\theta$

Equijoin

• A type of theta join where the predicate only consists of equalities.
• Example: $depositor \Join_{depositor.account_number = account.account_number} account$

Natural Join

• Notation: $r \Join s$
• Forms a cartesian product, selects tuples with matching values for attributes in $R \cap S$, and removes duplicate attributes in $S-R$.

Outer Join

• Retains information by adding tuples with null values when there are no matches.

Left Outer Join

• $r \Join^{\text{left}} s$

Right Outer Join

• $r \Join^{\text{right}} s$

Full Outer Join

• $r \Join^{\text{full}} s$

Division Operator

• Notation: $r \div s$
• Used for queries with "for all."
• $r \div s$ results in tuples of $r$ restricted to $R-S$.
• For every tuple in $s$, there is a tuple in $r$ with the same value as $t$ in attributes in $S$.
• Example: $\Pi_{customer_name, branch_name}(account \Join depositor) \div \Pi_{branch_name}(\sigma_{branch_city = "Brooklyn"}(branch))$ finds customers with accounts at all Brooklyn branches.

Lecture 24: Interference

Interference

• Waves interfere when overlapping in space.
• The resulting wave is the sum of individual waves and produces constructive or destructive outcomes

Constructive Interference

• Waves are in phase.
• The amplitude of the resultant wave exceeds that of each source wave.

Destructive Interference

• Waves are out of phase.
• The amplitude of the resultant wave is lower than that of each source wave.

Two-Source Interference

• Two wave sources are in phase.
• Path difference: $\Delta r = r_2 - r_1$
• Constructive interference: $\Delta r = m\lambda$, where $m = 0, 1, 2,...$
• Destructive interference: $\Delta r = (m + \frac{1}{2})\lambda$, where $m = 0, 1, 2,...$

Thin-Film Interference

• Light wave reflection happens on both top and bottom surfaces of a thin film, causing interference.
• Interference outcome depends on film thickness ($t$), refractive index ($n$), Light wavelength (${\lambda}$) and incidence angle

Phase Shift

• Reflection from a higher refractive index surface causes a $180^\circ$ (${\lambda/2}$) phase shift.
• No phase shift occurs when reflecting from a lower refractive index.

Constructive Interference Condition

• $2nt = m\lambda$, where $m = 0, 1, 2,...$

Destructive Interference Condition

• $2nt = (m + \frac{1}{2})\lambda$, where $m = 0, 1, 2,...$

Interference Example 1

• Oil film ($n = 1.45$) on water with strongest reflection at $520 \ nm$.
• Minimum thickness: $t = \frac{\lambda}{2n} = \frac{520 \ nm}{2(1.45)} = 179 \ nm$

Interference Example 2

• Magnesium fluoride ($MgF_2$) film on glass ($n = 1.38$) to reduce reflection at $550 \ nm$. Film's refractive index $n = 1.38$.
• Minimum thickness: $t = \frac{\lambda}{4n} = \frac{550 \ nm}{4(1.38)} = 99.6 \ nm$