UNIT 1 — Discrete Structures & Optimization Q1. Let p , q , r p,q,r be propositions. Which of the following is logically equivalent to p → ( q → r ) p \rightarrow (q \rightarrow r) ? A. ( p ∧ q ) → r (p\land q)\rightarrow r B. ( p ∨ q ) → r (p\lor q)\rightarrow r C. p → ( q ∧ r ) p\rightarrow(q\land r) D. ( p → q ) → r (p\rightarrow q)\rightarrow r Answer: A Explanation: p → ( q → r ) ≡ ¬ p ∨ ( ¬ q ∨ r ) ≡ ¬ ( p ∧ q ) ∨ r p\rightarrow(q\rightarrow r) \equiv \neg p\lor(\neg q\lor r) \equiv \neg(p\land q)\lor r Hence, ( p ∧ q ) → r (p\land q)\rightarrow r Q2. A connected undirected graph has exactly four vertices of odd degree. Which statement is necessarily true? A. It has an Euler circuit B. It has an Euler path but not an Euler circuit C. It has neither an Euler path nor an Euler circuit D. It must be a Hamiltonian graph Answer: C Explanation: An undirected connected graph has an Euler path iff it has exactly 0 or 2 odd-degree vertices . Four odd-degree vertices means neither Eul...
Connecting a laptop to a projector is usually straightforward: 1. Check the projector and laptop ports Common connections are: HDMI — best and most common VGA — older projectors/laptops USB-C — newer laptops; may need a USB-C-to-HDMI adapter 2. Connect the cable Turn on the projector. Connect the HDMI/VGA cable to the laptop. Connect the other end to the projector. On the projector, select the correct Input/Source (e.g., HDMI 1). 3. Select the display mode on Windows Press: Windows + P You'll see four options: PC screen only — projector off Duplicate — laptop and projector show the same screen ⭐ Extend — projector becomes a second monitor Second screen only — display only on projector For teaching/presentations, choose Duplicate . If the projector says "No Signal" Try these in order: Check that the projector's Source/Input matches the connected port. Unplug and reconnect the HDMI/VGA cable. Press Windows + P → Duplicate . Restart the projector. Try another HD...