Final answer:
1. Determine the truth values of compound statements. 2. Write contrapositive, inverse, and converse of conditional statements. 3. Show that (p ∧ -q) ∧ (p → q) is a contradiction. 4. Perform set operations on sets A and B.
Step-by-step explanation:
1. Truth Value of Compound Statements
A. Pq:
The compound statement Pq represents the conjunction of the two simple statements: p is an odd number and q is a prime number. To determine the truth value of Pq, we need to evaluate the truth values of p and q.
If p is true (15 is an odd number) and q is true (21 is a prime number), then Pq is true. Otherwise, if either p or q (or both) is false, then Pq is false.
B. -p-q:
The compound statement -p-q represents the negation of p followed by q. To determine the truth value of -p-q, we need to negate the truth value of p and then evaluate the truth value of q.
If p is true (15 is an odd number) and q is true (21 is a prime number), then -p-q is false. Otherwise, if either p or q (or both) is false, then -p-q is true.
2. Contrapositive, Inverse, and Converse of Conditional Statements
A. Conditional Statement: If it is cold, then the lake is frozen.
The contrapositive of a conditional statement swaps the hypothesis and conclusion, and negates both. In this case, the contrapositive is: If the lake is not frozen, then it is not cold.
The inverse of a conditional statement negates both the hypothesis and conclusion. In this case, the inverse is: If it is not cold, then the lake is not frozen.
The converse of a conditional statement swaps the hypothesis and conclusion. In this case, the converse is: If the lake is frozen, then it is cold.
3. Showing (p ∧ -q) ∧ (p → q) is a Contradiction
To show that (p ∧ -q) ∧ (p → q) is a contradiction, we need to show that it is always false, regardless of the truth values of p and q. We can do this by constructing a truth table.
If p is false and q is true, the statement becomes (false ∧ -true) ∧ (false → true), which simplifies to false ∧ true ∧ true, resulting in false.
If p is false and q is false, the statement becomes (false ∧ -false) ∧ (false → false), which simplifies to false ∧ true ∧ true, resulting in false.
If p is true and q is true, the statement becomes (true ∧ -true) ∧ (true → true), which simplifies to true ∧ false ∧ true, resulting in false.
If p is true and q is false, the statement becomes (true ∧ -false) ∧ (true → false), which simplifies to true ∧ true ∧ false, resulting in false.
Since the statement is always false, it is a contradiction.
4. Set Operations
I. If B is a subset of A, meaning every element of B is in A, then A must at least contain all the elements of B. If B = {6, 8}, then A must contain at least 6 and 8. Therefore, A={1,2,3,4,5,6,8}, which has a total of 7 elements.
II. To find A - B, we need to remove all the elements of B from A. Since B = {2, 5, 7, 8}, A - B = {1, 3, 4, 6}, which has a total of 4 elements.