Math 1103 Unit 1 Study Guide Review

Fundamentals of Logic: Conditional Statements and Variations

In logic, a conditional statement is often analyzed alongside its three variations: the converse, the inverse, and the contrapositive. These variations are formed by rearranging or negating the component statements $p$ (the antecedent) and $q$ (the consequent).

1. The Original Conditional: This is written as $p \rightarrow q$ (read as "if $p$, then $q$").

2. The Converse: This is formed by switching the antecedent and the consequent, written as $q \rightarrow p$ (read as "if $q$, then $p$").

3. The Inverse: This is formed by negating both the antecedent and the consequent, written as $\neg p \rightarrow \neg q$ (read as "if not $p$, then not $q$").

4. The Contrapositive: This is formed by switching and negating both components, written as $\neg q \rightarrow \neg p$ (read as "if not $q$, then not $p$").

The truth values for these four variations across all possible truth values of $p$ and $q$ are detailed in the following table:

Based on the observations from this truth table, logical equivalence can be determined. Two expressions are logically equivalent if their columns in a truth table are identical for every possible scenario. The original conditional ($p \rightarrow q$) is logically equivalent to the contrapositive ($\neg q \rightarrow \neg p$). Similarly, the converse ($q \rightarrow p$) and the inverse ($\neg p \rightarrow \neg q$) are logically equivalent to each other.

Verification of Logical Equivalence

To verify specific logical properties, such as the equivalence between a conditional and its algebraic expansion, truth tables are used. One common property is that a conditional statement $p \rightarrow q$ is equivalent to the disjunction of the negation of the antecedent and the consequent: $\neg p \lor q$.

Because the column for $p \rightarrow q$ is identical to the column for $\neg p \lor q$, the logical property is valid.

Symbolic Logic and Truth Value Evaluations

Compound logic involves evaluating the truth of statements based on simpler constituent statements or arithmetic facts. Consider the following simple statements and their evaluations:

Statement $\neg a$: $4 + 14 \neq 28$. Since the sum is $18$, and $18 \neq 28$, the statement is True.

Statement $e$: $9 + 7 = 17$. Since the sum is $16$, and $16 \neq 17$, the statement is False.

The disjunction $\neg a \lor e$ (read as "not $a$ or $e$") is evaluated as $\text{True} \lor \text{False}$. A disjunction is true if at least one of its components is true. Therefore, the truth value of the disjunction is True.

Translating English sentences into symbolic logic requires identifying the operators: "if… then" ($\rightarrow$), "if and only if" ($\leftrightarrow$), "and" ($\land$), "or" ($\lor$), and "not" ($\neg$). Using the variables $a$ (the lights are off), $b$ (it is bedtime), and $x$ (we go to sleep), complex sentences are translated as follows:

• "If it is bedtime, then the lights are off if and only if we go to sleep" translates to $b \rightarrow (a \leftrightarrow x)$.

• "If it is bedtime and we go to sleep, then the lights are off" translates to $(b \land x) \rightarrow a$.

• "It is bedtime if and only if the lights are off, and we go to sleep" translates to $(b \leftrightarrow a) \land x$.

• "If the lights are off is not true, then it is bedtime or we go to sleep" translates to $\neg a \rightarrow (b \lor x)$.

Set Theory: Operations and Properties

Set theory involves analyzing groups of objects within a Universal Set ($U$). Let $U = \{1, 2, 3, \dots, 20\}$. Specifically, we analyze subsets $A$ and $B$ defined as:

• $A = \{1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 15, 17, 18, 19\}$

• $B = \{2, 8, 9, 12, 13, 14, 16, 18, 19, 20\}$

To find the intersection of set $A$ and the complement of set $B$ ($A \cap B'$), we look for all elements that are in $A$ but NOT in $B$. Subtracting the common elements (2, 12, 18, 19) from $A$ results in: $A \cap B' = \{1, 3, 4, 5, 6, 7, 10, 11, 15, 17\}$

To find the intersection of set $B$ and the complement of set $A$ ($B \cap A'$), we look for elements in $B$ that are not present in $A$. Removing the intersection (2, 12, 18, 19) from $B$ results in: $B \cap A' = \{8, 9, 13, 14, 16, 20\}$

Venn diagrams help visualize operations involving multiple sets. For the negation of an intersection of three sets, $(A \cap B \cap C)'$, De Morgan's Law states that the complement of the intersection is the union of the individual complements. Mathematically, this is expressed as: $(A \cap B \cap C)' = A' \cup B' \cup C'$

Probability Theory and Spinner Experiments

Probability describes the likelihood of an outcome $s$ within a specific sample space. In experiments using spinners numbered sequentially,

Zero Probability: This occurs when an event is impossible within the given sample space. For a spinner numbered 1 to 7, the probability of spinning a 14 is zero, denoted as $P(s = 14) = 0$, because 14 is not in the set $(1, 2, 3, 4, 5, 6, 7)$.

Probability of One (Certainty): This occurs when every possible outcome in the sample space satisfies the condition. For a spinner numbered 1 to 10, the probability of spinning a number less than or equal to 10 is one ($P(s \le 10) = 1$).

Probability Between Zero and One: This describes events that are possible but not certain. For a spinner numbered 1 to 11, the probability of spinning a number greater than 10 is defined by the outcome {11}. Since only one outcome out of 11 satisfies the condition, the probability is P(s > 10) = \frac{1}{11}. This value ($0.0909\dots$) sits between 0 and 1.

Counting and Combinatorics in Configurations

When calculating the total number of possible configurations for a system with optional binary features (where each feature can either be included or not), the fundamental counting principle or powers of 2 can be applied.

If a car dealership offers $n$ optional features, such as a sunroof, navigation system, and heated seats ($n = 3$), each feature has 2 states (on or off). The number of total configurations is calculated as $2^n$: $2^3 = 2 \times 2 \times 2 = 8$ different possible car configurations.