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Course: LSAT > Unit 1

Lesson 3: Analytical Reasoning – Articles
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Analytical Reasoning – Articles
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Diagram notation conventions for analytical reasoning setups

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Diagram notation conventions for Analytical Reasoning setups: a quick guide

This article provides an inventory of the notations you will see throughout the Analytical Reasoning setups here at Khan Academy’s Official LSAT Practice. We consider these notations to be one set of “best practices”. That said, there is no “right way” of notating setups and their rules, and you should feel free to develop a system that works for you.
Whichever notations you wind up using, we strongly recommend that you arrive at a set of conventions that you like, and then practice using them efficiently and consistently as you prepare to do your best on Test Day.

Ordering notations

For the first two examples, we’ll show you the notation for a horizontal diagram on the left, and a vertical diagram on the right. For the remaining examples, we’ll show you how we’re notating the relationship in a horizontal ordering setup, and you can infer what the corresponding vertical-diagram notation would look like.
For more on ordering setups specifically, you can consult the article How to approach ordering setups.

A is before/above, but not necessarily immediately before/above, B.

Four chain diagrams include groupings of initials linked in sequence by a dotted or solid line. The chains read as follows. A is before dotted line B. A is before solid line B. A is above dotted line B. A is above solid line B.

A is immediately before/above B.

Two pairs of initials appear as follows. A before B is a box. A above B is a box.

A and B are a pair (next to each other), but we don’t know in what order.

A before B is a box. A double sided arrow extends from A to B above the box.

A is earlier than B, with at least one element between them.

A is before blank horizontal line, followed by dotted line B.

A is earlier than B, with exactly one element between them.

A is before blank horizontal line, followed by B.

A is sometime earlier than B, and A is also sometime earlier than C.

A chain diagram is a grouping of three initials linked in sequence by solid lines. The chain reads as follows. A is before B. A is before C.

A can never be immediately before B.

A before B is a box crossed out.

A and B can never be next to each other, regardless of the order.

A before B is a box crossed out. A double sided arrow extends from A to B above the box.

The same elements can never be consecutive.

X before X is a box crossed out.

There are least two spots, up to a maximum of four spots.

Two horizontal bars are followed by two horizontal dashed bars. The bars are blank and numbered from left to right as follows. 1, 2, 3, 4.

Placement, or assignment notations

A can never be in spot 1.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. Notations written below the bars read from left to right, by bar, as follows. Bar 1. A is crossed out.

Either A or B must be in spot 1.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. Bar entries read as follows from left to right. Bar 1. A, slash, B.

A must be in either spot 1 or 4.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. A is above the bars. An arrow extends from A to Bar 1. An arrow extends from A to Bar 4.

A must be in either spot 3 or 4.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. A is circled above Bars 3 and 4.

A and B are somewhere in spots 2, 3, and 4, but we don’t know which ones, and we don’t know in what order.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. A, comma, B is circled above Bars 2, 3 and 4.

A and B are a pair, and they are either in spots 2 and 3, or in spots 3 and 4, respectively.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. A before B is circled above Bars 2, 3 and 4.

A is sometime before B, though not necessarily immediately before B, and that relationship takes place somewhere between spots 2 and 4 inclusively.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. A, ellipsis, B is circled above Bars 2, 3 and 4.

We’ve tested out A in 3 and B in 4, but it breaks one of the other rules.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. Bar entries read as follows from left. Bar 3. A. Bar 4. B. The diagram is crossed out.

The same element must go into spots 1 and 4.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. Bar 1 includes a blank box. Bars 2 and 3 are blank. Bar 4 includes a blank box.

The elements in spots 1 and 4 must be different from each other.

Four horizontal bars are numbered from left to right as follows. 1, 2, 3, 4. Bar 1 includes a blank box. Bars 2 and 3 are blank. Bar 4 includes a blank triangle.

Grouping notations

For more on grouping setups specifically, you can consult the article How to approach grouping setups.

The elements will be grouped into categories X, Y, and Z, and we know that each category will have at least one element.

A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank.

The elements will be grouped into categories X, Y, and Z, and we know that each category will have at least one element, but that category X has only one element.

A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank. The line under column X is double ruled.

A can never be assigned to group X.

A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank. A notation under column X reads, A is crossed out.

Neither A nor B can be in group X, but they also can’t be a pair—in other words, one of A and B is in group Y, and the other of A and B is in group Z.

A grouping diagram includes 3 columns labeled from left to right as follows. x, y, z. Lines correspond to columns and read as follows. Column x is blank. Column y, A slash B. Column z, A slash B.

A and B are a pair—in other words, they’re both in the same category/group.

A above B is a box.

A and B cannot be a pair—in other words, they can’t be in the same category.

A above B is a box crossed out.

If A is in group X, then B must be in group Z.

A diagram includes a grouping diagram and notations. A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank. Notations read as follows. If, A sub x, right arrow, B sub z. Notation 2 reads as follows. If, B sub z is crossed out, right arrow, A sub x is crossed out.

There are fewer elements in group X than there are in group Z.

A diagram includes a grouping diagram and a notation. A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank. A notation reads as follows. X is less than Z.

There are fewer of element A present than there are of element B, but there's at least one of each type of element.

A diagram includes a grouping diagram and a notation. A grouping diagram includes 3 columns labeled from left to right as follows. X, Y, Z. Below each column is a corresponding line. All lines are blank. A notation reads as follows. A is less than B.

Conditional notations

Note: the second notation in each rule is a logically equivalent version of the stated rule (also known as the
). For more on logically equivalent rules, you can consult the article Conditional reasoning and logical equivalence.

Conditional rules about groups

Examples:
  • If A is selected, B is also selected.
  • If A is in a category, B must also be in that category.
  • A is in a category only if B is in that category.
  • A cannot be selected unless B is also selected.
  • B must be selected in order for A to be selected.
Notations read as follows. If A, right arrow, B. If B is crossed out, right arrow, A is crossed out.

Conditional rules about placement

  • If A is in spot 2, then B must be in spot 3.
  • A can be in spot 2 only if B is in spot 3.
  • A can’t be in spot 2 unless B is in spot 3.
  • B must be in spot 3 in order for A to be in spot 2.
Notations read as follows. If A 2, right arrow, B 3. If B 3 is crossed out, right arrow, A 2 is crossed out.

Conditional rules about quantity

  • If two A elements are present, then three B elements must also be present.
  • The only way that two A elements can be selected is if three B elements are selected.
  • Two A elements are present only if three B elements are present.
  • Three B elements must be selected in order for two A elements to be selected.
  • Two A elements cannot be present unless three B elements are present.
Notations read as follows. If 2 A, right arrow, 3 B. If 3 B is crossed out, right arrow, 2 A is crossed out.

"If and only if"

  • A is selected if and only if B is selected.
Notations read as follows. If A, double sided arrow, B. If B is crossed out, double arrow, A is crossed out.

Conditional rules with AND/OR

  • If A is selected, then B cannot be selected and C cannot be selected (or, if A is selected, then neither B nor C can be selected).
  • In setups with categories, if A is part of a category, then neither B nor C can be part of that category.
  • We can infer that AB will never be a pair, and AC will never be a pair.
Notations read as follows. If A, right arrow, B is crossed out, and, C is crossed out. If B or C, right arrow, A is crossed out.

Conditional rules with mixed notation

  • If A is earlier than B, then C cannot be in spot 3.
  • The logically equivalent rule is that if C is in spot 3, then either B is before A, or B and A are at the same time.
Notations read as follows. If A, ellipsis, B, right arrow, C3 is crossed out. If C3, right arrow, A, ellipsis, B is crossed out.

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  • marcimus pink style avatar for user Princess Sandy
    Posted 6 years ago. Direct link to Princess Sandy's post “Are these the same notati...”
    Are these the same notations they teach in law school?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Dasha S
    Posted 4 years ago. Direct link to Dasha S's post “There appears to be a typ...”
    There appears to be a typo in the last note of the "If and only if" section, namely the "if [A]" in "in setups with categories, this notation means that if A is part of a category if and only if B is also part of that category."
    (3 votes)
    Default Khan Academy avatar avatar for user
  • purple pi pink style avatar for user ACh1
    Posted 2 years ago. Direct link to ACh1's post “Under "Conditional rules ...”
    Under "Conditional rules with AND/OR," it says "We can infer that AB will never be a pair, and AC will never be a pair." But couldn't they technically be a pair if they were both not in the selected category (not A & not B or not A & not C)?
    (2 votes)
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    • blobby green style avatar for user siyangleoliu
      Posted 2 years ago. Direct link to siyangleoliu's post “I agree with you. o If Ar...”
      I agree with you.
      o If Art is selected, then neither Biology nor Chemistry can be selected.
      o If Art is selected, then Biology cannot be selected and Chemistry cannot be selected.
      o We can infer that Art and Biology will never be a pair, OR Art and Chemistry will never be a pair.

      If Biology is selected, then Art is definitely not selected. However, chemistry doesn't conflict with any rules. Chemistry could be selected or not be selected. If B => -A, then we cannot deduct anything about C. Assume there are only two categories, C could be in the same category as B, or C could be in the same category as A. C cannot be decided to be or not to be with A.

      The same applies if C => -A, then B cannot be decided to be or not to be with A.
      (1 vote)
  • blobby green style avatar for user Jamie Parris
    Posted a year ago. Direct link to Jamie Parris 's post “When you have a rule that...”
    When you have a rule that is a is before b, it can be written like a...b. How many options can be between them? If you worked out a problem and you put a and b next to each other, is that breaking a rule? Would that make them a pair? Thanks! :D
    (1 vote)
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