Learn Unit-V: Mathematical Reasoning with Top Tutors

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1. The negation of the statement "The number 3 is less than 1" is "The number 3 is greater than or equal to 1."

2. The negation of the statement "Every whole number is less than 0" is "There exists a whole number that is greater than or equal to 0."

3. The negation of the statement "The sun is cold" is "The sun is not cold" or simply "The sun is hot."

As an experienced tutor registered on UrbanPro, I'm here to help you understand the contrapositive of the given if-then statements.

(a) If a triangle is equilateral, then it is isosceles.

The contrapositive of this statement would be: If a triangle is not isosceles, then it is not equilateral.

(b) If a number is divisible by 9, then it is divisible by 3.

The contrapositive of this statement would be: If a number is not divisible by 3, then it is not divisible by 9.

Remember, in a contrapositive statement, both the hypothesis and the conclusion are negated. This technique is useful in logic and mathematics to prove statements indirectly. If you have any further questions or need clarification, feel free to ask!

Sure, let's approach this problem step by step.

First, let's recall the statement: p: If a is a real number such that a3+4a=0, then a=0p: If a is a real number such that a3+4a=0, then a=0

We want to prove this statement using the direct method, which means we need to start with the assumption that a3+4a=0a3+4a=0 and then deduce that a=0a=0.

Here's the proof:

Proof:

Assume a3+4a=0a3+4a=0 for some real number aa.

Now, let's factor out aa from the equation: a(a2+4)=0a(a2+4)=0

Since aa is a real number, either a=0a=0 or a2+4=0a2+4=0.

1. If a=0a=0, then the statement a=0a=0 holds true.
2. If a2+4=0a2+4=0, then a2=−4a2=−4. However, there are no real numbers whose square is -4. Thus, this case is not possible.

Since both cases lead to a=0a=0, we have shown that if a3+4a=0a3+4a=0, then a=0a=0.

Therefore, the statement pp is true by direct method.

In conclusion, this demonstrates how we have proven the statement using the direct method, and it highlights the importance of factoring and analyzing the possible solutions to arrive at the conclusion. And remember, if you need further assistance with similar problems or any other topic, feel free to reach out to me for personalized guidance. Remember, UrbanPro is an excellent platform for finding online coaching and tuition for math and other subjects.