Unit 3: Mathematical Reasoning

Mathematical Reasoning involves taking a given set of facts and using logic to draw conclusions. These logical frameworks are critical for algorithm design, data analysis, and general problem-solving.

1. Odd Man Out

Odd Man Out is the process of identifying a given series or group of elements and locating the one element that doesn’t share the common characteristics of the rest.

Key Strategies

• Look for mathematical patterns (primes, squares, cubes, multiples).
• Check alphabetical patterns (vowels/consonants, skip sequences).
• Observe physical or conceptual attributes.

Example: Find the odd one out: $5, 7, 11, 13, 15, 17$ Solution: $15$. All other numbers are prime numbers, whereas $15$ is a composite number ($3 \times 5$).

2. Syllogism

A Syllogism is a form of deductive reasoning where a conclusion is drawn from two or more given premises. Venn diagrams are frequently used to solve syllogisms accurately.

Example Premises:

1. All computers are machines.
2. All machines require power.

Conclusion: Therefore, all computers require power.

Common Patterns

• All A are B: Represents A as a subset of B.
• Some A are B: Represents an intersection between A and B.
• No A are B: Represents disjoint sets A and B.

3. Blood Relations

Blood Relation problems test the ability to understand genealogical structures and the relationships between family members based on specific clues.

Family Tree Mapping

To accurately represent relations, construct a family tree:

• Use horizontal lines — for siblings.
• Use vertical lines | for parent-child relationships.
• Use a + sign for male, and - sign for female (or squares/circles).
• Use a = sign for a married couple.

Example Problem: A says, “B is the father of my sister’s son.” How is B related to A? Solution: A’s sister’s son = A’s nephew. The father of A’s nephew is A’s sister’s husband. Therefore, B is A’s brother-in-law.

4. Coding Decoding

Coding is a method of encrypting a word or number based on a specific set of rules. Decoding is the process of reverting the encrypted format back to the original based on the same rules.

Common Techniques

• Letter Shifting: Shifting each letter forward or backward by a specific value (e.g., A $\to$ C, B $\to$ D).
• Position Value: Mapping letters to their numerical positions in the alphabet (A=1, B=2, Z=26) and applying arithmetic operations.
• Substitution: Replacing a word with another word or a symbol unconditionally.

Example Problem: If CPU is coded as D QV, how will RAM be coded? Solution: C + 1 = D P + 1 = Q U + 1 = V Therefore, applying the same logic: R + 1 = S, A + 1 = B, M + 1 = N. The code for RAM is SBN.

Competency-Based Questions

1. (Odd Man Out): Considering binary patterns in computer science, which of these numbers is the odd man out in decimal form: $2, 4, 8, 16, 24, 32$? Justify your reasoning mathematically.

2. (Syllogism Analysis): Premise 1: No algorithms are hardware. Premise 2: All CPUs are hardware. Draw a Venn diagram to determine if the conclusion “No CPUs are algorithms” is logically valid.

3. (Blood Relations Logic): In a database representing a digital family tree structure, Entity $X$ is logged as the mother of the father of Entity $Y$’s sister. How is Entity $X$ related to Entity $Y$?

4. (Coding / Cryptography): A basic encryption algorithm shifts the letters of the alphabet backward by 3 places (i.e., D becomes A, E becomes B, etc.). An intercepted network packet contains the ciphertext ZHOFRPH. Decode this ciphertext to reveal the original message.

5. (Composite Reasoning): A company employs 5 people: A, B, C, D, E. B is the brother of A. C is the father of B. D is the sister of E. E is the daughter of A. Based on these relationships, create a family tree and determine how D is related to C. Additionally, if you were to assign sequential employee IDs based on age (oldest first), what logical assumptions must you make?

Answers to Competency-Based Questions

1. Odd Man Out: The sequence is ${2, 4, 8, 16, 24, 32}$. Mathematically, $2=2^1$, $4=2^2$, $8=2^3$, $16=2^4$, and $32=2^5$. All these are pure powers of $2$. $24$ is mathematically $8 \times 3$, which is not a pure power of $2$. Therefore, 24 is the odd man out.

2. Syllogism Analysis: Let $U$ be the universal set of all objects.

• $H$ = set of Hardware.
• $A$ = set of Algorithms.
• $C$ = set of CPUs. Premises state: $A \cap H = \emptyset$ (No algorithms are hardware) and $C \subseteq H$ (All CPUs are hardware). Since all elements of $C$ are in $H$, and $H$ shares no elements with $A$, it must be that $C$ shares no elements with $A$ ($C \cap A = \emptyset$). Thus, “No CPUs are algorithms” is logically valid.

3. Blood Relations Logic:

• Entity $Y$’s sister shares the same father as $Y$. Let this father be $F$.
• The mother of $F$ is logged as Entity $X$.
• Since $F$ is the father of $Y$, $X$ is the mother of $Y$’s father. Therefore, $X$ is the paternal grandmother of $Y$.

4. Coding / Cryptography: If shifting backward by 3 places encrypts the data (meaning Decryption requires a forward shift of +3), then treating ZHOFRPH with +3 yields CKRIUSK, which has no meaning. Instead, observe standard Caeser shifts: the ciphertext ZHOFRPH when shifted backward by 3 places (Z-3=W, H-3=E, O-3=L, F-3=C, R-3=O, P-3=M, H-3=E) yields WELCOME. Thus, the encryption shifted the original message forward, and the decoding rule is a backward shift of 3. Code: WELCOME.

5. Composite Reasoning:

• $C$ is the father of $B$. $B$ is the brother of $A$. So, $C$ is the father of both $A$ and $B$.
• $E$ is the daughter of $A$. $D$ is the sister of $E$. So, $D$ is the daughter of $A$.
• Since $D$ is the daughter of $A$, and $C$ is the father of $A$, $D$ is the granddaughter of $C$.
• Assumptions for Age IDs: To assign age-based IDs, you must logically assume that parents are strictly older than their children ($C > A, B$ and $A > D, E$). You must also assume or require external data to tie-break between siblings ($A$ vs $B$, and $D$ vs $E$), as the family tree alone does not determine sibling seniority.