Q. A positive plus a positive equals.

Q: Can a positive plus a positive equal zero?

No. If a > 0 and b > 0 then a + b > 0. Zero would require at least one nonpositive addend, so two positives cannot sum to zero.

Q: Is a+b always greater than each addend when a,b>0?

Yes. For strictly positive a and b, a+b>a and a+b>b because you are adding a strictly positive quantity to each addend.

Q: What is the smallest value a+b can take if a,b>0 are real?

What is the smallest value a+b can take if a,b>0 are real?

Q: Does addition of positive numbers preserve inequalities (monotonicity)?

Yes. If a>c and b>0 then a+b>c+b. Adding the same positive amount to two numbers preserves and strictens inequalities.

Answer

Let \(a>0\) and \(b>0\). Since both are positive, their sum is positive, because adding positive numbers gives a number greater than 0. Thus \(a+b>0\).

Detailed Explanation

Problem

Show that a positive plus a positive is positive.

Solution (step-by-step, with detailed explanation)

State the definitions and assumptions.

Let \(a\) and \(b\) be real numbers and assume they are positive. By definition of positive, this means

\(a > 0\) and \(b > 0\).
Use the property that adding the same real number to both sides of a strict inequality preserves the inequality.

Starting from \(a > 0\), add \(b\) to both sides. The justification is: if \(x > y\) then for any real \(z\) we have \(x + z > y + z\). Applying this with \(x = a\), \(y = 0\), and \(z = b\) yields

\(a + b > 0 + b\).

Since \(0 + b = b\), this simplifies to

\(a + b > b\).
Use transitivity of the strict order.

We have \(a + b > b\) from the previous step and we also have \(b > 0\) by assumption. The relation \(>\) is transitive: if \(X > Y\) and \(Y > Z\) then \(X > Z\). Applying transitivity with \(X = a + b\), \(Y = b\), and \(Z = 0\) gives

\(a + b > 0\).
Conclude the result.

Therefore the sum of two positive real numbers is positive. In formula form:

\( \text{If } a > 0 \text{ and } b > 0 \text{ then } a + b > 0.\)

Final answer: \(a + b > 0\).

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FAQs

If \(a>0\) and \(b>0\), is \(a+b>0\)?

Yes. The sum of two strictly positive numbers is strictly positive because adding positive quantities increases value: \(a+b > a > 0\) and \(a+b > b > 0\).

Can a positive plus a positive equal zero?

No. If \(a > 0\) and \(b > 0\) then \(a + b > 0\). Zero would require at least one nonpositive addend, so two positives cannot sum to zero.

If \(a\) and \(b\) are positive real numbers, is \(a+b\) a positive real?

Yes. The positive real numbers are closed under addition, so for \(a, b \in \mathbb{R}\) with \(a>0, b>0\), we have \(a+b \in \mathbb{R}\) and \(a+b>0\).

If \(a\) and \(b\) are positive integers, what can you say about \(a+b\)?

The sum is a positive integer. The smallest possible sum is \(1+1=2\). So \(a+b\in\mathbb{Z}^+\) and \(a+b\ge 2\).

Is \(a+b\) always greater than each addend when \(a,b>0\)?

Yes. For strictly positive \(a\) and \(b\), \(a+b>a\) and \(a+b>b\) because you are adding a strictly positive quantity to each addend.

What is the smallest value \(a+b\) can take if \(a,b>0\) are real?

Does addition of positive numbers preserve inequalities (monotonicity)?

Yes. If \(a>c\) and \(b>0\) then \(a+b>c+b\). Adding the same positive amount to two numbers preserves and strictens inequalities.

Are properties like commutativity and associativity valid for positive numbers under addition?

Yes. For positives, as for all real numbers, addition is commutative and associative: \(a+b=b+a\) and \((a+b)+c=a+(b+c)\) for any positive \(a,b,c\).

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