T07 · B2 — Conditionals and Branch Selection

5. The identity comparison. Confirmed twice: (a+b+c) > (c+b+a) is always false — both sides are equal and > is strict.

5. (a) 44
| Start | a=10 | b=4 | c=6 | | |---|---|---|---|---| | a>b | true | | | | | b=a+b | a=10 | b=14 | c=6 | | | b>a | 14>10 true | | | | | c=c+b | a=10 | b=14 | c=20 | final=44 | - (b) 20: treats outer a>b false, no changes.
- (c) 34: uses old b=4 in c=c+b, giving c=10.
- (d) 40: treats inner b>a false, executes c=c+a, giving c=16.

6. Truthiness of integers. A condition like ((r*q)+q) is true whenever it is non-zero.

No pen and paper. Hold three variables as a mental triple and restate them after every assignment.

3. Concept notes

3.1 Boolean foundations

In these papers a condition is any expression that evaluates to a truth value. The rule is:

So a condition like if ((r * q) + q) is not a comparison — it is a truthiness test.

The −1, 5 case is the trap: non-zero is not the same as positive.

Truth tables

3.2 Relational operators

The single most important fact: > is strictly greater. If both sides are equal, the result is FALSE.

Worked example:

a = 4
b = 9
c = 2
if ((a + b + c) > (c + b + a)) then
    print "YES"
else
    print "NO"
end if

Both sides are 4 + 9 + 2 = 15. The test is 15 > 15, which is FALSE. Output: NO.

Any comparison of an expression with itself reordered is always FALSE for > and <, and always TRUE for >=, <=, ==.

3.3 Branch constructs

Generic syntax:

if (condition) then
    statements
end if

if (condition) then
    statements
else
    statements
end if

if (c1) then
    s1
else if (c2) then
    s2
else if (c3) then
    s3
else
    s4
end if

if (outer) then
    if (inner) then
        s
    end if
end if

Ladder vs nested: in a ladder, at most one branch executes and the rest are never tested. In a nested if, both outer and inner must be TRUE for the innermost body to run, and tests happen top-down.

Matching End if: If and End if pair like brackets. The nearest unmatched If is closed by the next End if (innermost-first). Indentation mirrors this:

if (A) then          <- If #1
    if (B) then      <- If #2
        x = 1
    end if           <- closes If #2
    y = 2            <- still inside If #1
end if               <- closes If #1

A line indented at the level of If #1 is outside If #2, so it runs whenever the outer guard is TRUE — even when the inner guard is FALSE. This is exactly Trap 2 below.

switch / case

switch (n)
    case 1:
        x = x + 1
        break
    case 2:
        x = x + 2
        break
    default:
        x = 0
end switch

Fall-through worked example:

x = 0
switch (2)
    case 1: x = x + 1
    case 2: x = x + 2
    case 3: x = x + 3
    default: x = x + 10
end switch
print x

Output 15. With a break after each case, the output would be 2. If no case matches and there is no default, nothing executes.

3.4 Dry-run / trace table technique

One column per variable, one row per executed line. Add a "Note" column for the condition result. Update a variable's cell only on the line that assigns it; carry the old value forward otherwise.

Snippet:

1  a = 3
2  b = 4
3  c = a * b
4  if (c > 10) then
5      a = c - b
6  end if
print a, b, c

Output: 8 4 12.

Rules that make this fast:

• Write the initial values in row 0 before reading any condition.
• Only assignment lines change a cell.
• print reads the current cell values, i.e. the last row.

3.5 Short-circuit evaluation

Why it changes output: the skipped operand never runs, so any side effect inside it never happens, and any error inside it never occurs.

a = 0
b = 5
if (a != 0 AND b / a > 1) then
    print "A"
else
    print "B"
end if

a != 0 is FALSE, so b / a is never evaluated (no division by zero) and the AND is FALSE. Output B.

count = 0
flag = 0
if (flag != 0 OR count = count + 1) then
    print "X"
end if

flag != 0 is FALSE, so the OR does evaluate the second operand and count becomes 1.

Fast eliminator: if neither operand has a side effect and neither can error, short-circuit cannot change the printed output. Ignore it and evaluate normally.

3.6 The four branch-selection traps

Trap 1 — FALSE outer guard makes the body dead code

x = 5
y = 3
if (x < y) then
    x = x + 10
    y = y * 2
end if
print x, y

Output: 5 3.

Rule: evaluate the outermost condition BEFORE tracing the body. If it is FALSE, the answer is the current variable values. This alone answers many questions in seconds.

Trap 2 — outer TRUE but inner FALSE

a = 4
b = 9
if (a < b) then
    if (a > b) then
        a = a + b
        b = a - b
    end if
end if
print a, b

Output: 4 9.

Wrong answer to avoid: if the inner guard were TRUE the body would give a = 13, then b = 13 − 9 = 4. Candidates who assume "we entered the outer block, so the body runs" answer 13 4. It is 4 9.

Trap 3 — a variable reassigned before a later condition reads it

a = 10
b = 7
if (a > b) then
    a = a - 5
end if
if (a > b) then
    b = a * 2
else
    b = b * 2
end if
print a, b

Correct trace (second condition uses the NEW a):

Output: 5 14.

Stale-value trace (using the original a = 10 at line 6):

Wrong answer: 10 20. The branch flips and the value is wrong. Whenever a variable is reassigned above, cross out its old value.

Trap 4 — the always-false identity comparison

x = 6
y = 2
z = 9
if ((x + y + z) > (z + y + x)) then
    print "YES"
else
    print "NO"
end if

Output: NO. Both sides are the same three variables summed, so > is FALSE and >=/<=/== would be TRUE.

3.7 Operator precedence inside a condition

Highest binds first:

Arithmetic binds tighter than relational, which binds tighter than logical. So:

• a + b > c → (a + b) > c
• a > b AND c > d → (a > b) AND (c > d)
• NOT a > b → NOT (a > b) — NOT sits below relational, above AND
• a OR b AND c → a OR (b AND c)

Use brackets when tracing to remove all doubt.

3.8 60-second attack checklist

4. Must-memorise block

Truth tables

Truthy / falsy

if ((r * q) + q) is TRUE whenever (r*q)+q ≠ 0. With r = −1, q = 5 it is 0 → FALSE.

Relational operators

Strict-greater rule: if both sides are equal, > is FALSE and < is FALSE.

Always-false identity pattern: (a + b + c) > (c + b + a) is always FALSE, whatever the values. Same expression reordered is never greater than itself. (>=, <=, == are always TRUE.)

The four traps

Branch constructs

End-if matching: pair If/End if like brackets, innermost first. A line indented at the outer level is outside the inner If and runs whenever the outer guard is TRUE.

Short-circuit

Matters only when the skipped operand has a side effect or can error (e.g. division by zero). Otherwise it cannot change the output.

Precedence (high → low): ( ) → * / % → + - → < <= > >= == != → NOT → AND → OR. Arithmetic beats relational; relational beats logical.

60-second attack checklist

1. Write all initial values.
2. Apply assignments that sit above the first condition.
3. Evaluate the outermost condition. FALSE → dead code → skip to the print.
4. TRUE → test each inner condition separately.
5. Use reassigned values in every later condition.
6. Equal sides with > or < → FALSE, instantly.

6. (b) 13
| Start | a=5 | b=8 | c=2 | | |---|---|---|---|---| | a<b | true | | | | | b=b-a | a=5 | b=3 | c=2 | | | b<c | 3<2 false | | | | | c=c+b | a=5 | b=3 | c=5 | final=13 | - (a) 12: treats inner b<c true, executes a=a+c, giving a=7.
- (c) 15: treats outer a<b false, no changes.
- (d) 23: does not update b before inner test; c=c+b uses b=8, giving c=10.

7. switch: check break (fall-through) and default.

7. (a) 28
| Start | a=2 | b=6 | c=3 | | |---|---|---|---|---| | a28 false | | | final=28 | - (b) 29: treats identity guard as true, increments a.
- (c) 11: treats outer a<b false, no changes.
- (d) 18: skips c=a+b but executes a=c+b using old c=3, giving a=9.

1  a = 4
2  b = 7
3  c = 2
4  if a > b
5      c = a + b
6      if c > 10
7          a = c - b
8      else
9          b = c + a
10 Print a + b + c

1  a = 6
2  b = 8
3  c = 3
4  if a < b
5      if c > a
6          a = a + c
7          b = b - c
8  Print a + b + c

1  a = 3
2  b = 9
3  c = 5
4  if a < b
5      c = a + b
6      if c > b
7          a = c + b
8      else
9          b = c - a
10 Print a + b + c

1  a = 2
2  b = 4
3  c = 6
4  if (a + b + c) > (c + b + a)
5      a = a + 10
6      b = b + 10
7      c = c + 10
8  if a < b
9      c = a + b + 5
10 Print a + b + c

1  a = 10
2  b = 4
3  c = 7
4  if a > b
5      b = a - c
6      if b > c
7          a = a - b
8      else
9          c = b + a
10 Print a + b + c

1 a = 3
2 b = 5
3 c = 7
4 if a > b then
5   a = a + 10
6   b = b + 10
7   c = c + 10
8 end if
9 Print a + b + c

1 a = 2
2 b = 9
3 c = 4
4 if (a + b + c) > (c + b + a) then
5   a = a + 10
6   b = b + 10
7   c = c + 10
8 end if
9 Print a + b + c

1 a = 6
2 b = 2
3 c = 9
4 if a > b then
5   if c < b then
6     a = a + 1
7     b = b + 1
8     c = c + 1
9   end if
10 end if
11 Print a + b + c

1 a = 4
2 b = 7
3 c = 3
4 if a < b then
5   a = a + 5
6   if a > b then
7     c = c + a
8   end if
9 end if
10 Print a + b + c

1 a = 10
2 b = 4
3 c = 6
4 if a > b then
5   b = a + b
6   if b > a then
7     c = c + b
8   else
9     c = c + a
10  end if
11 end if
12 Print a + b + c

1 a = 5
2 b = 8
3 c = 2
4 if a < b then
5   b = b - a
6   if b < c then
7     a = a + c
8   else
9     c = c + b
10  end if
11 end if
12 Print a + b + c

1 a = 2
2 b = 6
3 c = 3
4 if a < b then
5   c = a + b
6   a = c + b
7 end if
8 if (a + b + c) > (c + b + a) then
9   a = a + 1
10 end if
11 Print a + b + c

1  a = 3
2  b = 4
3  c = 5
4  IF a > b THEN
5      b = a + b
6      c = b + a
7  END IF
8  Print a + b + c

1  p = 6
2  q = 2
3  r = 9
4  IF p > q THEN
5      IF r < p THEN
6          q = p + r
7          r = q - p
8      END IF
9  END IF
10 Print p + q + r

1  a = 5
2  b = 9
3  c = 2
4  a = a + b
5  IF a > 10 THEN
6      c = a - b
7  ELSE
8      c = b - a
9  END IF
10 Print a + b + c

1  a = 7
2  b = 3
3  c = 6
4  IF (a + b + c) > (c + b + a) THEN
5      a = a + b + c
6      b = a - c
7  ELSE
8      c = a + b
9  END IF
10 Print a + b + c

1  p = 10
2  q = 4
3  r = 7
4  IF p > q THEN
5      r = p - q
6  END IF
7  IF r > p THEN
8      p = r + q
9  ELSE
10     q = r + p
11 END IF
12 Print p + q + r

1  a = 8
2  b = 5
3  c = 3
4  IF (a + b + c) > (c + b + a) THEN
5      a = a + 1
6  ELSE IF a > b + c THEN
7      c = a + b
8  ELSE
9      c = b + c
10 END IF
11 Print a + b + c