Introduction to Logic: Mathematical Literacy (Coursework Sample)

Introduction to Logic (Mathematical Literacy)
1 Hypothesis and conclusion
LOGIC, we could say, is the study of If − then sentences.
If a number is divisible by 10, then it is divisible by 2.
The clause introduced by If –
A number is divisible by 10
– is called the hypothesis.
It is what we are given, or what we may assume. The clause introduced by then –
It is divisible by 2
–is called the conclusion.
☞ Conclusion is the statement that “follows” from the hypothesis. Or, given the hypoth- esis, it is the statement that we must prove.
2 Necessary and sufficient conditions
Sufficient Condition: When the If −then sentence is true, we say that the hypothesis is a sufficient condition for the conclusion.
Thus it is sufficient to know that a number is divisible by 10 – in order to conclude that it is divisible by 2.
Necessary Condition: The conclusion is then called a necessary condition of that hypothesis.
Because it necessarily follows that the number will be divisible by 2, if it is divisible by 10.
Remark. The hypothesis is a sentence, independent of “If”, and the conclusion is a sentence, independent of “then.”
Problem 1. If two numbers are even, then their sum is even.
1 Name the hypothesis and the conclusion.
2 Which of them is the sufficient condition, and which the necessary condition?
Answer.(i). Hypothesis: Two numbers are even.
Conclusion: Their sum is even.
(ii). Since the statement is true, we have, Sufficient Condition: Two numbers are even. Necessary Condition: Their sum is even.
Problem 2. A number is divisible by 3 if it is divisible by 6.
Name the hypothesis and the conclusion. Which of them is the sufficient condition, and which the necessary condition?
Answer. The hypothesis is “A number is divisible by 6.” It is that clause which follows “if.” It is the sufficient condition, because the statement is true.
The conclusion is “The number is divisible by 3.” It is the necessary condition.
Problem 3. If a number ends in 0, then it is a multiple of 5.
* Name the hypothesis and the conclusion.
* Is the hypothesis a sufficient condition for that conclusion?
Answer.(i). Hypothesis: A number ends in 0.
Conclusion: It is a multiple of 5.
(ii). Yes, because the statement is true.
Problem 4. If a number is a multiple of 5, then it ends in 0.
* Name the hypothesis and the conclusion.
* Is the hypothesis a sufficient condition for that conclusion?
Answer.(i). Hypothesis: A number is a multiple of 5.
Conclusion: It ends in 0.
(ii). No, because the statement is false. 15 is a multiple of 5, but 15 does not end in 0.