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Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\qquad \textbf{(E)}\ \text{For some x}, x^2 \... | 1. **Identify the original statement and its structure**: The original statement is "For all $x$ of a certain set, $x^2 > 0$". This is a universal quantification statement, which asserts that a certain property (in this case, $x^2 > 0$) holds for every element $x$ in a specified set.
2. **Understand the negation of a ... | [
"<think>\nOkay, so I need to find the negation of the statement: \"For all x of a certain set, x squared is greater than zero.\" Hmm, negation... Let me think. \n\nFirst, I remember that the negation of a universal statement (\"for all\") is an existential statement (\"there exists\") with the negation of the inner... | 10 | [
0,
1,
2
] | All three solutions agree on the final answer being E, thus they all share the same answer. | [
"<think>\nOkay, so I need to find the negation of the statement: \"For all x of a certain set, x squared is greater than zero.\" Hmm, negation... Let me think. \n\nFirst, I remember that the negation of a universal statement (\"for all\") is an existential statement (\"there exists\") with the negation of the inner... | <think>
Okay, so I need to find the negation of the statement: "For all x of a certain set, x squared is greater than zero." Hmm, negation... Let me think.
First, I remember that the negation of a universal statement ("for all") is an existential statement ("there exists") with the negation of the inner statement. So... | [
{
"from": "human",
"value": "Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?\n$\\textbf{(A)}\\ \\text{For all x}, x^2 < 0\\qquad \\textbf{(B)}\\ \\text{For all x}, x^2 \\le 0\\qquad \\textbf{(C)}\\ \\text{For no x}, x^2>0\\qquad \\\\ \\textbf{(D)}\\ \\text{For... |
Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\qquad \textbf{(E)}\ \text{For some x}, x^2 \... | 1. **Identify the original statement and its structure**: The original statement is "For all $x$ of a certain set, $x^2 > 0$". This is a universal quantification statement, which asserts that a certain property (in this case, $x^2 > 0$) holds for every element $x$ in a specified set.
2. **Understand the negation of a ... | [
"<think>\nOkay, so I need to find the negation of the statement: \"For all x of a certain set, x squared is greater than zero.\" Hmm, negation... Let me think. \n\nFirst, I remember that the negation of a universal statement (\"for all\") is an existential statement (\"there exists\") with the negation of the inner... | 10 | [
0,
1,
2
] | All three solutions agree on the final answer being E, thus they all share the same answer. | [
"<think>\nOkay, so I need to find the negation of the statement: \"For all x of a certain set, x squared is greater than zero.\" Hmm, negation... Let me think. \n\nFirst, I remember that the negation of a universal statement (\"for all\") is an existential statement (\"there exists\") with the negation of the inner... | <think>
Alright, so I have this problem here that asks for the negation of a statement. The original statement is: "For all x of a certain set, x² > 0." And then I have five options to choose from.
Hmm, okay, so first off, I remember that when you negate a statement that starts with "for all," it becomes "there exists... | [
{
"from": "human",
"value": "Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?\n$\\textbf{(A)}\\ \\text{For all x}, x^2 < 0\\qquad \\textbf{(B)}\\ \\text{For all x}, x^2 \\le 0\\qquad \\textbf{(C)}\\ \\text{For no x}, x^2>0\\qquad \\\\ \\textbf{(D)}\\ \\text{For... |
Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\qquad \textbf{(E)}\ \text{For some x}, x^2 \... | 1. **Identify the original statement and its structure**: The original statement is "For all $x$ of a certain set, $x^2 > 0$". This is a universal quantification statement, which asserts that a certain property (in this case, $x^2 > 0$) holds for every element $x$ in a specified set.
2. **Understand the negation of a ... | [
"<think>\nOkay, so I need to find the negation of the statement: \"For all x of a certain set, x squared is greater than zero.\" Hmm, negation... Let me think. \n\nFirst, I remember that the negation of a universal statement (\"for all\") is an existential statement (\"there exists\") with the negation of the inner... | 10 | [
0,
1,
2
] | All three solutions agree on the final answer being E, thus they all share the same answer. | [
"<think>\nOkay, so I need to find the negation of the statement: \"For all x of a certain set, x squared is greater than zero.\" Hmm, negation... Let me think. \n\nFirst, I remember that the negation of a universal statement (\"for all\") is an existential statement (\"there exists\") with the negation of the inner... | <think>
Okay, so I have this problem here where I need to find the negation of the statement: "For all x of a certain set, x squared is greater than zero." The options are labeled from A to E, and I need to figure out which one is the correct negation. Let me try to break it down step by step.
First, I should recall h... | [
{
"from": "human",
"value": "Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?\n$\\textbf{(A)}\\ \\text{For all x}, x^2 < 0\\qquad \\textbf{(B)}\\ \\text{For all x}, x^2 \\le 0\\qquad \\textbf{(C)}\\ \\text{For no x}, x^2>0\\qquad \\\\ \\textbf{(D)}\\ \\text{For... |
"Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not always correct is:\n$\\text(...TRUNCATED) | "We are given that $x > y$ and both $x$ and $y$ are positive. We need to determine which of the give(...TRUNCATED) | ["<think>\nAlright, so I need to figure out which inequality isn't always correct given the conditio(...TRUNCATED) | 4 | [
0,
1,
2
] | All three solutions agree that the inequality which is not always correct is C. | ["<think>\nAlright, so I need to figure out which inequality isn't always correct given the conditio(...TRUNCATED) | "<think>\nAlright, so I need to figure out which inequality isn't always correct given the condition(...TRUNCATED) | [{"from":"human","value":"Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not al(...TRUNCATED) |
"Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not always correct is:\n$\\text(...TRUNCATED) | "We are given that $x > y$ and both $x$ and $y$ are positive. We need to determine which of the give(...TRUNCATED) | ["<think>\nAlright, so I need to figure out which inequality isn't always correct given the conditio(...TRUNCATED) | 4 | [
0,
1,
2
] | All three solutions agree that the inequality which is not always correct is C. | ["<think>\nAlright, so I need to figure out which inequality isn't always correct given the conditio(...TRUNCATED) | "<think>\nOkay, so I need to figure out which inequality isn't always correct given the conditions. (...TRUNCATED) | [{"from":"human","value":"Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not al(...TRUNCATED) |
"Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not always correct is:\n$\\text(...TRUNCATED) | "We are given that $x > y$ and both $x$ and $y$ are positive. We need to determine which of the give(...TRUNCATED) | ["<think>\nAlright, so I need to figure out which inequality isn't always correct given the conditio(...TRUNCATED) | 4 | [
0,
1,
2
] | All three solutions agree that the inequality which is not always correct is C. | ["<think>\nAlright, so I need to figure out which inequality isn't always correct given the conditio(...TRUNCATED) | "<think>\nOkay, let's see. I have this problem where I need to figure out which inequality isn't alw(...TRUNCATED) | [{"from":"human","value":"Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not al(...TRUNCATED) |
"At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. Ther(...TRUNCATED) | "1. **Identify the number of students in each class:**\n - Mrs. Germain's class: 11 students\n -(...TRUNCATED) | ["<think>\nTo determine the total number of mathematics students at Euclid Middle School taking the (...TRUNCATED) | 0 | [
0,
1,
2
] | "All three solutions state that the total number of students taking the AMC 8 contest is 28. Therefo(...TRUNCATED) | ["<think>\nTo determine the total number of mathematics students at Euclid Middle School taking the (...TRUNCATED) | "<think>\nTo determine the total number of mathematics students at Euclid Middle School taking the A(...TRUNCATED) | [{"from":"human","value":"At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. New(...TRUNCATED) |
"At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. Ther(...TRUNCATED) | "1. **Identify the number of students in each class:**\n - Mrs. Germain's class: 11 students\n -(...TRUNCATED) | ["<think>\nTo determine the total number of mathematics students at Euclid Middle School taking the (...TRUNCATED) | 0 | [
0,
1,
2
] | "All three solutions state that the total number of students taking the AMC 8 contest is 28. Therefo(...TRUNCATED) | ["<think>\nTo determine the total number of mathematics students at Euclid Middle School taking the (...TRUNCATED) | "<think>\nFirst, note the number of students in each mathematics teacher's AMC 8 class:\n- Mrs. Germ(...TRUNCATED) | [{"from":"human","value":"At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. New(...TRUNCATED) |
"At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. Ther(...TRUNCATED) | "1. **Identify the number of students in each class:**\n - Mrs. Germain's class: 11 students\n -(...TRUNCATED) | ["<think>\nTo determine the total number of mathematics students at Euclid Middle School taking the (...TRUNCATED) | 0 | [
0,
1,
2
] | "All three solutions state that the total number of students taking the AMC 8 contest is 28. Therefo(...TRUNCATED) | ["<think>\nTo determine the total number of mathematics students at Euclid Middle School taking the (...TRUNCATED) | "<think>\nTo determine the total number of mathematics students at Euclid Middle School taking the A(...TRUNCATED) | [{"from":"human","value":"At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. New(...TRUNCATED) |
"What time was it $2011$ minutes after midnight on January 1, 2011? \n$\\textbf{(A)}\\ \\text{Januar(...TRUNCATED) | "1. **Convert minutes to hours and minutes**: \n We start by converting the total minutes (2011) i(...TRUNCATED) | ["<think>\nFirst, I'll convert the total minutes into hours and minutes. There are 60 minutes in an (...TRUNCATED) | 9 | [
0,
1,
2
] | All three solutions conclude with the same final answer: January 2 at 9:31 AM. | ["<think>\nFirst, I'll convert the total minutes into hours and minutes. There are 60 minutes in an (...TRUNCATED) | "<think>\nFirst, I'll convert the total minutes into hours and minutes. There are 60 minutes in an h(...TRUNCATED) | [{"from":"human","value":"What time was it $2011$ minutes after midnight on January 1, 2011? \n$\\te(...TRUNCATED) |
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