1.a. 123five
+ 34five
b. 1010two
- 101two
c. 23five
* 34five
c. What is the contrapositive of r → q?
d. The statements p → q and q → r are given. If we know q is true, is p necessarily true? Explain.
a. To add numbers written in base five, you need to carry over if the sum of the two digits in the same place value is five or greater.
In this case, you have:
1 + 3 = 4 (base five) in the units place.
2 + 4 = 11 (base five). You carry over one to the fives place and write 1 in the units place.
1 + 0 + 3 = 4 (base five) in the fives place.
So, the sum of 123five and 34five is 441five.
b. To subtract numbers written in base two, you compare the digits in the same place value. If the digit of the first number is greater than or equal to the digit of the second number, you subtract normally. If the digit of the first number is smaller, you borrow from the next higher place value.
In this case, you have:
1 - 1 = 0 in the ones place.
0 - 0 = 0 in the twos place.
1 - 1 = 0 in the fours place.
1 - 0 = 1 in the eights place.
So, the difference between 1010two and 101two is 1000two.
c. To multiply numbers written in base five, you multiply each digit of the first number by each digit of the second number, starting from the rightmost digit. Then you add the results, carrying over as necessary.
In this case, you have:
3 * 4 = 12 (base five) in the units place.
2 * 4 = 8 (base five) in the fives place. You carry over one to the twenty-fives place and write 2 in the fives place.
1 * 4 = 4 (base five) in the twenty-fives place.
So, the product of 23five and 34five is 424five.
d. The contrapositive of the implication "r -> q" is "not q -> not r". In other words, if q is false, then r must also be false.
To determine if p is necessarily true given that q is true, we need to consider the truth values of p and r. If p is true, then the implication "p -> q" is satisfied regardless of the truth value of r. However, if p is false, then the truth value of "p -> q" depends on the truth value of q.
Since we are given that q is true, p can be either true or false, and the implication "p -> q" is always satisfied. Therefore, p is not necessarily true when q is true.
Explanation: In an implication of the form "p -> q," the truth of the conclusion q does not imply the truth of the hypothesis p.