Math / Discrete

QuestionPart I (10 pt., 2 pt. each) Indicate whether each of the following statements is true or false.
1. $|A|=|B|$ where $A=\{1,2,\{3,4\}\}$ and $B=\{a, a,\{a\},\{a, b\}\}$ .
2. $\{3,4\} \subseteq A$ where $A=\{1,2,\{3,4\}\}$ .
3. $\{3,4\} \in P(A)$ where $A=\{1,2,\{3,4\}\}$ and $P(A)$ is the power set of $A$ .
4. $(A \cup \emptyset) \cap \emptyset=A$ where $A$ is a set.
5. $\sum_{i=1}^{10} i=\sum_{j=10}^{20}(j-10)$

Studdy Solution

STEP 1

1. The cardinality of a set $|A|$ is the number of distinct elements in the set.
2. The symbol $\subseteq$ denotes a subset, meaning all elements of the first set are contained in the second set.
3. The symbol $\in$ denotes membership, meaning the element is contained within the set.
4. The power set $P(A)$ is the set of all subsets of $A$ .
5. The union of a set with the empty set is the set itself.
6. The intersection of any set with the empty set is the empty set.
7. The summation notation $\sum$ denotes the sum of a sequence of numbers.

STEP 2

1. Determine the truth value of statement 1.
2. Determine the truth value of statement 2.
3. Determine the truth value of statement 3.
4. Determine the truth value of statement 4.
5. Determine the truth value of statement 5.

STEP 3

Determine the truth value of statement 1: $|A| = |B|$ .
- Set $A = \{1, 2, \{3, 4\}\}$ has 3 distinct elements: $1$ , $2$ , and $\{3, 4\}$ . - Set $B = \{a, a, \{a\}, \{a, b\}\}$ has 3 distinct elements: $a$ , $\{a\}$ , and $\{a, b\}$ (note that repeated elements do not count multiple times in a set).
Therefore, $|A| = |B| = 3$ .
Statement 1 is \textbf{True}.

STEP 4

Determine the truth value of statement 2: $\{3,4\} \subseteq A$ .
- The set $A = \{1, 2, \{3, 4\}\}$ contains the element $\{3, 4\}$ as a single element, not as separate elements $3$ and $4$ .
Therefore, $\{3, 4\} \subseteq A$ is \textbf{False} because $\{3, 4\}$ is not a subset of the individual elements of $A$ .

STEP 5

Determine the truth value of statement 3: $\{3,4\} \in P(A)$ .
- The power set $P(A)$ includes all subsets of $A$ . - Since $\{3, 4\}$ is an element of $A$ , it is a subset of $A$ .
Therefore, $\{3, 4\} \in P(A)$ is \textbf{True}.

STEP 6

Determine the truth value of statement 4: $(A \cup \emptyset) \cap \emptyset = A$ .
- The union of any set $A$ with the empty set $\emptyset$ is $A$ . - The intersection of any set with the empty set is $\emptyset$ .
Therefore, $(A \cup \emptyset) \cap \emptyset = \emptyset$ , not $A$ .
Statement 4 is \textbf{False}.

STEP 7

Determine the truth value of statement 5: $\sum_{i=1}^{10} i = \sum_{j=10}^{20}(j-10)$ .
- Calculate $\sum_{i=1}^{10} i = 1 + 2 + \ldots + 10 = \frac{10 \times 11}{2} = 55$ . - Calculate $\sum_{j=10}^{20}(j-10) = (10-10) + (11-10) + \ldots + (20-10) = 0 + 1 + \ldots + 10 = \frac{10 \times 11}{2} = 55$ .
Both sums equal 55.
Statement 5 is \textbf{True}.
The truth values of the statements are:
1. True
2. False
3. True
4. False
5. True

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