docs: 添加离散数学理论基础 2024 春夏第三次小测试题

Author: Xecades

docs/major_basic/discrete_math/Discrete_Mathematics_Quiz_3_2024.pdf

@@ -0,0 +1,172 @@
+#import "@preview/numblex:0.1.1": numblex
+#import "@preview/diagraph:0.2.1": raw-render
+#import "@preview/cetz:0.2.2"
+
+#set text(font: "Times New Roman", size: 11pt)
+#set par(leading: 1.1em, justify: true)
+#set enum(numbering: numblex(numberings: ("1.", "(a)")), full: true, spacing: 2em)
+#set figure(supplement: "Fig.", gap: 15pt, caption: "")
+#set figure.caption(separator: "")
+
+#let und(w: 5em) = box(width: w, line(length: 100%, stroke: .5pt))
+#let header-fn-sized = size => it => [
+    #set align(center)
+    #set text(size: size, font: "FZXiaoBiaoSong-B05S")
+    #it.body
+]
+#let graph = x => figure(cetz.canvas(x))
+#let node = (coord, name) => {
+    import cetz.draw: *
+    circle(coord, name: name, radius: .3)
+    content(name, name)
+}
+#let raw_edge = (u, v, w, marked: false) => {
+    import cetz.draw: *
+    set-style(content: (frame: "rect", stroke: none, fill: white, padding: .05))
+    if marked { set-style(mark: (end: "straight")) }
+    let name = "edge_" + u + "_" + v
+    line(u, v, name: name)
+    content(name + ".mid", [#w])
+}
+#let edge = raw_edge.with(marked: false)
+#let dedge = raw_edge.with(marked: true)
+#let redge = cetz.draw.line
+
+#show heading.where(level: 1): header-fn-sized(20pt)
+#show heading.where(level: 2): header-fn-sized(13pt)
+#show heading.where(level: 3): header-fn-sized(13pt)
+
+#show regex("(\d+%)"): set text(style: "italic")
+
+= Discrete Mathematics Quiz 3
+
+== 2023-2024 春夏学期
+
+=== Xecades
+
+#v(2em)
+
++ $R={(a,a), (a,b), (b,d), (a,d)}$ is a relation on ${a, b, c, d}$. Find the smallest relation containing the relation $R$ that is:
+  + (6%) partial order relation.
+  + (6%) symmetric and transitive.
+
++ Given the undirected graph $G$ as shown in @fig1.
+  + (6%) Use Kruskal's algorithm to find the minimun spanning tree of graph $G$. What is the order in which the edges are added to the minimum spanning tree?
+    #graph({
+        node((0, 0), "c")
+        node((3, 0), "d")
+        node((0, 3), "a")
+        node((3, 3), "b")
+        node((1.5, 1.5), "e")
+        node((4.5, 1.5), "f")
+        edge("a", "b", 20)
+        edge("a", "c", 12)
+        edge("a", "e", 9)
+        edge("b", "e", 11)
+        edge("b", "d", 6)
+        edge("b", "f", 5)
+        edge("c", "e", 10)
+        edge("c", "d", 18)
+        edge("d", "e", 14)
+        edge("d", "f", 7)
+    })<fig1>
+  + (6%) Using alphabetical ordering, find a spanning tree for this graph by depth-first search.
+
++ (6%) The frequencies of six characters are $0.09$, $0.05$, $0.2$, $0.25$, $0.3$ and $0.11$, respectively. If Huffman coding is used for optimal encoding, the average number of bits required to encode a character is #und().
+
++ (6%) How many leaves does a full $7$-ary tree with $2024$ vertices have?
+
++ (6%) Determine all positive integers $r$ and $s$ for which the complete bipartite graph $K_(r,s)$ is a tree.
+
++ (6%) Suppose $abs(A)=4$. Find the number of different equivalence relations on $A$.
+
++ Answer these questions for the poset $({2, 3, 5, 6, 12, 20, 27, 36, 60}, |)$.
+  + (4%) Draw the Hasse diagram.
+  + (2%) Find the maximal elements.
+  + (2%) Is there a least element?
+  + (2%) Find all upper bound of ${2, 3}$.
+
++ (10%) In the network below (@fig2), find a maximum flow from $A$ to $J$, calculate its flow value, and prove that it is the maximum flow.
+    #graph({
+        node((0, 1.5), "G")
+        node((0, 3), "D")
+        node((0, 4.5), "B")
+        node((4, 1.5), "H")
+        node((8, 1.5), "I")
+        node((8, 0), "J")
+        node((6, 3), "F")
+        node((6, 4.5), "C")
+        node((3, 6), "A")
+        dedge("B", "D", 10)
+        dedge("D", "G", 2)
+        dedge("D", "H", 9)
+        dedge("G", "H", 7)
+        dedge("H", "I", 2)
+        dedge("G", "J", 9)
+        dedge("H", "J", 9)
+        dedge("I", "J", 4)
+        dedge("F", "H", 3)
+        dedge("F", "I", 3)
+        dedge("B", "F", 2)
+        dedge("A", "B", 13)
+        dedge("A", "C", 7)
+        dedge("B", "C", 7)
+        dedge("C", "F", 9)
+    })<fig2>
+
++ (8%) Determine if the given pair of graphs (@fig3) is isomorphic. Give the reason.
+    #figure(grid(
+        columns: 2,
+        column-gutter: 2em,
+        cetz.canvas({
+            node((0, 0), "7")
+            node((0, 1), "5")
+            node((0, 2), "3")
+            node((0, 3), "1")
+            node((2, 0), "8")
+            node((2, 1), "6")
+            node((2, 2), "4")
+            node((2, 3), "2")
+            redge("1", "2")
+            redge("1", "4")
+            redge("1", "6")
+            redge("3", "2")
+            redge("3", "4")
+            redge("3", "8")
+            redge("5", "2")
+            redge("5", "6")
+            redge("5", "8")
+            redge("7", "4")
+            redge("7", "6")
+            redge("7", "8")
+        }),
+        cetz.canvas({
+            node((0, 0), "g")
+            node((3, 0), "h")
+            node((0, 3), "a")
+            node((3, 3), "b")
+            node((1, 1), "e")
+            node((2, 1), "f")
+            node((1, 2), "c")
+            node((2, 2), "d")
+            redge("a", "b")
+            redge("b", "h")
+            redge("h", "g")
+            redge("g", "a")
+            redge("c", "d")
+            redge("d", "f")
+            redge("f", "e")
+            redge("e", "c")
+            redge("a", "c")
+            redge("b", "d")
+            redge("h", "f")
+            redge("g", "e")
+        })
+    ))<fig3>
+
++ $Q_n$ is the graph with $2^n$ vertices representing bit strings of length $n$. An edge exists between two vertices that differ in exactly one bit position.
+  + (3%) Find the number of edges of $Q_5$.
+  + (3%) Find the chromatic number of $Q_5$. Give the reason.
+  + (6%) Determing is $Q_5$ has Hamilton circuit / path. Give the reason.
+
++ (12%) $8$ students take a test with $8$ true / false questions. It is known that no two students make exactly the same choice. Prove that we can remove one of the $8$ questions, and still no two students make exactly the same choice.

docs/major_basic/discrete_math/Discrete_Mathematics_Quiz_3_2024.typ

@@ -49,6 +49,7 @@
 - [（22 级）2023 春夏第一次小测](Discrete_Mathematics_Quiz_1_2023.pdf)
 - [（22 级）2023 春夏第二次小测](Discrete_Mathematics_Quiz_2_2023.pdf)
 - [（23 级）2024 春夏第一次小测](Discrete_Mathematics_Quiz_1_2024.pdf)
+- [（23 级）2024 春夏第三次小测](Discrete_Mathematics_Quiz_3_2024.pdf)
 
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