Question

1. p ∨ q: atlanta is the capital of georgia and there are 13 inches in a foot.

p ∧ q: atlanta is the capital of georgia and there are 13 inches in a foot.

1. p ∧ ¬q: atlanta is the capital of georgia and there are not 13 inches in a foot.
2. ¬p ∨ q: atlanta is not the capital of georgia and there are 13 inches in a foot.

determine if the conjectures below are true or false. if false, give a counterexample.

1. for any two integers, a and b, (a - b)^2 = a^2 - b^2
2. the difference of two odd numbers is always an odd number.
3. two right angles are always supplementary.

Explanation:

Step1: Expand left - hand side of equation

Expand \((a - b)^2\) using the formula \((x - y)^2=x^{2}-2xy + y^{2}\). Here \(x = a\) and \(y = b\), so \((a - b)^2=a^{2}-2ab + b^{2}\).

Step2: Compare with right - hand side

Since \(a^{2}-2ab + b^{2}
eq a^{2}-b^{2}\) for non - zero \(b\). Let \(a = 2\) and \(b = 1\). Then \((a - b)^2=(2 - 1)^2=1\), and \(a^{2}-b^{2}=2^{2}-1^{2}=4 - 1 = 3\).

Step3: Analyze statement about odd numbers

Let two odd numbers be \(2m+1\) and \(2n + 1\), where \(m\) and \(n\) are integers. Their difference is \((2m + 1)-(2n+1)=2m + 1-2n - 1=2(m - n)\), which is an even number. For example, if \(m = 2\) and \(n = 1\), the odd numbers are \(2\times2+1 = 5\) and \(2\times1+1 = 3\), and \(5 - 3=2\) (even).

Step4: Analyze statement about right angles

A right angle measures \(90^{\circ}\). The sum of two right angles is \(90^{\circ}+90^{\circ}=180^{\circ}\). By the definition of supplementary angles (two angles whose sum is \(180^{\circ}\)), two right angles are supplementary.

Answer:

1. False. Counter - example: \(a = 2\), \(b = 1\), \((a - b)^2 = 1\), \(a^{2}-b^{2}=3\).
2. False. Counter - example: Let the odd numbers be 5 and 3, \(5 - 3 = 2\) (even).
3. True.