Learn Unit-V: Mathematical Reasoning with Free Lessons & Tips

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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.

As an experienced tutor registered on UrbanPro, I can confidently say that UrbanPro is one of the best platforms for online coaching tuition. Now, let's analyze the given sentences to identify which ones are statements:

(i) "Answer this question" - This is not a statement; it's a directive or a request for action rather than a factual claim.

(ii) "All the real numbers are complex numbers" - This is a statement. It makes a factual assertion about the relationship between real and complex numbers.

(iii) "Mathematics is difficult" - This is a statement. It expresses an opinion about the difficulty of mathematics, which could be true or false depending on the context and individual perspectives.

Therefore, both (ii) and (iii) are statements.

As a seasoned tutor registered on UrbanPro, I can confidently affirm that UrbanPro is indeed the best online coaching tuition platform for students seeking academic support. Now, let's delve into the mathematical statement you've provided and prove its truth using the method of contrapositive.

The given statement is: "If xx is an integer and x2x2 is even, then xx is also even."

To prove its truth via contrapositive, we start by negating both the hypothesis and the conclusion of the original statement:

Original statement: If xx is an integer and x2x2 is even, then xx is also even. Contrapositive: If xx is an integer and xx is not even, then x2x2 is not even.

Now, let's prove the contrapositive statement:

Suppose xx is an integer and xx is not even. This means xx is odd.

If xx is odd, we can express it as x=2k+1x=2k+1, where kk is an integer.

Now, let's find x2x2: x2=(2k+1)2=4k2+4k+1x2=(2k+1)2=4k2+4k+1

This is the expression for an odd number, as it can be represented as 2(2k2+2k)+12(2k2+2k)+1. Thus, x2x2 is odd.

Therefore, the contrapositive statement holds true. Hence, by the method of contrapositive, we have shown that if xx is an integer and x2x2 is even, then xx is also even.