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24:[["$","$L33",null,{"feature":"flashcards"}],"$L34",["$","$L35",null,{"href":"/gcse/maths/aqa/grade-5-5-23-venn-diagrams/flashcards","label":"Venn Diagrams","subjectCode":"gcse_maths__aqa","surface":"flashcards"}],["$","$L36",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L37",null,{"decks":[{"id":"ebc2b9e8-2368-d9f2-a61e-926620d3cb81","title":"Venn Diagrams Flashcards","topicId":"929a3f0c-f3a9-45f0-9f12-edba92b20630","topicSlug":"grade-5-5-23-venn-diagrams","cards":[{"id":"b4ab0f03-21b0-ab70-ed86-a201307c0779","cardType":"basic","points":1,"front":"In frequency notation, what does $n(A)$ represent?","back":"The number of elements in set $A$.","mcqOptions":[{"text":"The number of elements in set $A$.","correct":true,"explanation":"The notation $n(A)$ gives the cardinality of $A$, meaning the total number of distinct elements contained in that set."},{"text":"The number of elements outside set $A$.","correct":false,"explanation":"This confuses a set with its complement. Elements outside $A$ are counted by $n(A')$, while $n(A)$ counts the elements inside $A$."},{"text":"The number of subsets formed from set $A$.","correct":false,"explanation":"The number of subsets of a set with $n(A)$ elements is $2^{n(A)}$. The notation $n(A)$ itself counts the elements in $A$."},{"text":"The probability of selecting set $A$.","correct":false,"explanation":"Probability is written as $P(A)$, not $n(A)$. The notation $n(A)$ represents the number of elements contained in set $A$."}]},{"id":"2e003666-dcf6-cfbd-3ce5-a4b0dbabedb7","cardType":"basic","points":2,"front":"What symbol is used for the universal set and how is it represented on a diagram?","back":"

Symbol: $\\mathcal{E}$
Representation: The rectangle

","mcqOptions":[{"text":"

Symbol: $\\mathcal{E}$
Representation: The rectangle

","correct":true,"explanation":"The universal set is denoted by $\\mathcal{E}$ in this notation and is drawn as the rectangle containing every set and all relevant elements."},{"text":"

Symbol: $A$
Representation: A circle

","correct":false,"explanation":"A labelled circle represents an individual set such as $A$, not the universal set. The universal set is $\\mathcal{E}$ and is represented by the rectangle."},{"text":"

Symbol: $\\varnothing$
Representation: An empty circle

","correct":false,"explanation":"The symbol $\\varnothing$ denotes the empty set, which has no elements. The universal set is denoted by $\\mathcal{E}$ and shown as the surrounding rectangle."},{"text":"

Symbol: $A \\cap B$
Representation: The overlap

","correct":false,"explanation":"The notation $A \\cap B$ and the overlap represent the intersection of two sets. The universal set is $\\mathcal{E}$ and is drawn as the rectangle."}]},{"id":"174ddabc-0af3-125c-9969-82329e560d7c","cardType":"cloze","points":null,"front":"The complement $A'$ represents all elements in the universal set that are [ ].","back":"The complement $A'$ represents all elements in the universal set that are not in A.","mcqOptions":[{"text":"not in A","correct":true,"explanation":"The complement of $A$ contains every element belonging to the universal set that does not belong to $A$."},{"text":"in A","correct":false,"explanation":"This describes $A$ itself rather than its complement. The complement $A'$ consists of elements in the universal set that are not in $A$."},{"text":"in every set","correct":false,"explanation":"A complement is not defined by membership of every set. The set $A'$ contains precisely the universal-set elements that are not in $A$."},{"text":"outside the universal set","correct":false,"explanation":"A complement only includes elements within the universal set. Specifically, $A'$ contains those universal-set elements that are not in $A$."}]},{"id":"0b48bf33-f553-8529-1e6a-dfe87b32f902","cardType":"basic","points":1,"front":"Define the intersection symbol $\\cap$.","back":"Items that are in both set A and set B (the overlap).","mcqOptions":[{"text":"Items that are in both set A and set B (the overlap).","correct":true,"explanation":"The intersection $A \\cap B$ consists of elements shared by A and B, so it is represented by the overlapping region of their circles."},{"text":"Items that are in either set A or set B (the union).","correct":false,"explanation":"This describes the union $A \\cup B$, which includes elements in either set. The intersection $A \\cap B$ contains only elements in both sets."},{"text":"Items that are in only set A and not set B (A-only).","correct":false,"explanation":"The A-only region excludes the overlap and is written $A \\cap B'$. The intersection $A \\cap B$ is the region belonging to both sets."},{"text":"Items that are in neither set A nor set B (the outside).","correct":false,"explanation":"The region outside both sets is the complement of their union, $(A \\cup B)'$. The intersection $A \\cap B$ is their shared overlap."}]},{"id":"3233ad3f-bebb-3800-6b8c-88f3ed9cbd55","cardType":"basic","points":1,"front":"Define the union symbol $\\cup$.","back":"Items in A or B or both.","mcqOptions":[{"text":"Items in A or B or both.","correct":true,"explanation":"The union $A \\cup B$ contains every element belonging to A, to B, or to both sets, so both circles are included."},{"text":"Items in A and B at once.","correct":false,"explanation":"This describes the intersection $A \\cap B$, which includes only the overlap. The union $A \\cup B$ includes items in A, B, or both."},{"text":"Items in A but not B at all.","correct":false,"explanation":"This is the A-only region, written $A \\cap B'$. The union $A \\cup B$ includes all of A and B, including their overlap."},{"text":"Items outside A and B entirely.","correct":false,"explanation":"This describes elements in neither set, written $(A \\cup B)'$. The union $A \\cup B$ contains elements in A, B, or both."}]},{"id":"d387342a-782b-902c-18f3-63178bf1af5e","cardType":"cloze","points":null,"front":"$$$P(A') = \\text{[...]}$$","back":"$$$P(A') = \\boxed{1 - P(A)}$$","mcqOptions":[{"text":"$$$1 - P(A)$$","correct":true,"explanation":"An event and its complement cover all possible outcomes, so their probabilities total 1. Therefore, $P(A')=1-P(A)$."},{"text":"$$$1 + P(A)$$","correct":false,"explanation":"Adding produces a value greater than 1 whenever $P(A)>0$, which cannot be a probability. The complement rule is $P(A')=1-P(A)$."},{"text":"$$$P(A)$$","correct":false,"explanation":"An event and its complement do not generally have equal probabilities; that occurs only when each is 0.5. In general, $P(A')=1-P(A)$."},{"text":"$$$\\frac{1}{P(A)}$$","correct":false,"explanation":"Taking a reciprocal is not the complement rule and often gives a value above 1. Complementary probabilities sum to 1, so $P(A')=1-P(A)$."}]},{"id":"fe33621c-72e6-727a-fdb4-e10c2009b8f9","cardType":"basic","points":1,"front":"If $P(A) = 0.64$, what is $P(A')$?","back":"$$$1 - 0.64 = 0.36$$","mcqOptions":[{"text":"$$$1 - 0.64 = 0.36$$","correct":true,"explanation":"Complementary probabilities add to 1, so subtracting $0.64$ from 1 gives $P(A')=0.36$."},{"text":"$$$1 - 0.64 = 0.64$$","correct":false,"explanation":"This leaves the original probability unchanged instead of carrying out the subtraction. The complement is $1-0.64=0.36$."},{"text":"$$$1 - 0.64 = 0.46$$","correct":false,"explanation":"This reverses the digits in the correct difference. Subtracting carefully gives $1.00-0.64=0.36$, so $P(A')=0.36$."},{"text":"$$$1 + 0.64 = 1.64$$","correct":false,"explanation":"This adds the probability instead of subtracting it, producing an impossible probability above 1. The complement is $1-0.64=0.36$."}]},{"id":"d8914d25-2af9-8eef-44ea-e7a365a3b14e","cardType":"basic","points":1,"front":"Which region corresponds to the notation $A \\cap B'$?","back":"The A-only region (in A but not in B).","mcqOptions":[{"text":"The A-only region (in A but not in B).","correct":true,"explanation":"The intersection with $B'$ selects elements that satisfy both conditions: they belong to A and do not belong to B."},{"text":"The B-only region (in B but not in A).","correct":false,"explanation":"The B-only region is $A' \\cap B$, with the complement applied to A. The notation $A \\cap B'$ instead describes the A-only region."},{"text":"The overlap region (in both A and B).","correct":false,"explanation":"The overlap is $A \\cap B$. The prime in $B'$ excludes elements in B, so $A \\cap B'$ is the A-only region."},{"text":"The outside region (in neither A nor B).","correct":false,"explanation":"The region in neither set is $A' \\cap B'$, or $(A \\cup B)'$. The notation $A \\cap B'$ requires membership of A."}]},{"id":"a166e148-f7a6-1cbb-1d01-b732a8650140","cardType":"cloze","points":null,"front":"The region for \"neither A nor B\" is represented by $A' \\cap B'$, which is located [ ].","back":"The region for \"neither A nor B\" is represented by $A' \\cap B'$, which is located outside both circles.","mcqOptions":[{"text":"outside both circles","correct":true,"explanation":"By De Morgan’s law, $A' \\cap B'$ contains elements not in A and not in B, so the required region lies outside both circles."},{"text":"inside both circles","correct":false,"explanation":"Inside both circles represents $A \\cap B$, where elements belong to both sets. Neither set instead means $A' \\cap B'$, which is outside both circles."},{"text":"inside circle A only","correct":false,"explanation":"The A-only region contains elements in A but not B, represented by $A \\cap B'$. Neither A nor B lies outside both circles."},{"text":"inside at least one circle","correct":false,"explanation":"Inside at least one circle represents the union $A \\cup B$. Neither A nor B is the complement of this union, located outside both circles."}]},{"id":"3fc94acf-fec6-7296-49b4-fdfc587417ea","cardType":"basic","points":2,"front":"What is the rule for completing a frequency Venn diagram when the overlap is known?","back":"Fill the overlap first, then subtract it from the circle totals to find the \"only\" regions.","mcqOptions":[{"text":"Fill the overlap first, then subtract it from the circle totals to find the \"only\" regions.","correct":true,"explanation":"The overlap contributes to both circle totals. Entering it first allows it to be subtracted from each total, leaving the correct counts for the two \"only\" regions."},{"text":"Fill the circle totals first, then add the overlap to find the \"only\" regions.","correct":false,"explanation":"Adding the overlap double-counts elements belonging to both sets. The overlap must be filled first and subtracted from each circle total to obtain the \"only\" regions."},{"text":"Fill the only regions first, then subtract them from the overlap to find the circle totals.","correct":false,"explanation":"The \"only\" regions cannot be determined directly while the known overlap is unused. Fill the overlap first, then subtract it from each circle total."},{"text":"Fill the outside region first, then subtract it from the circle totals to find the \"only\" regions.","correct":false,"explanation":"The outside count is not included in either circle total, so subtracting it is inappropriate. Fill the known overlap first and subtract that from the circle totals."}]},{"id":"d8753624-5e6c-aecd-73f2-85ce8f12631a","cardType":"basic","points":2,"front":"How do you find the count of the overlap if given circle totals and the union total?","back":"Add the two circle totals together and subtract the total count of the union.","mcqOptions":[{"text":"Add the two circle totals together and subtract the total count of the union.","correct":true,"explanation":"Adding the circle totals counts the overlap twice, while the union counts it once. Subtracting the union total therefore leaves one copy of the overlap."},{"text":"Add the two circle totals together and add the total count of the union.","correct":false,"explanation":"Adding the union creates extensive double-counting rather than isolating the overlap. Add the circle totals, then subtract the union total to leave the overlap count."},{"text":"Subtract the two circle totals from each other and add the total count of the union.","correct":false,"explanation":"The difference between circle totals does not isolate shared elements. Inclusion-exclusion gives the overlap by adding both circle totals and subtracting the union total."},{"text":"Subtract the total count of the union and subtract the two circle totals together.","correct":false,"explanation":"Subtracting all the given counts cannot recover the shared portion and may produce a negative result. Add the circle totals, then subtract the union total."}]},{"id":"bd018cde-c83f-9dc1-2928-8d4a01ecb644","cardType":"basic","points":3,"front":"State the order for filling a three-circle Venn diagram from a survey.","back":"

Centre (all three sets)
Pair-only overlaps
Single-only regions
Outside region

","mcqOptions":[{"text":"

Centre (all three sets)
Pair-only overlaps
Single-only regions
Outside region

","correct":true,"explanation":"The centre affects every pair and circle total, so it is entered first. Successive subtraction then determines pair-only, single-only, and finally outside regions."},{"text":"

Single-only regions
Pair-only overlaps
Centre (all three sets)
Outside region

","correct":false,"explanation":"Starting with single-only regions ignores values that must first be removed from circle totals. Begin with the centre, then pair-only overlaps, single-only regions, and outside."},{"text":"

Outside region
Single-only regions
Pair-only overlaps
Centre (all three sets)

","correct":false,"explanation":"Working from outside inward prevents the necessary subtractions from known intersections. Fill the centre first, followed by pair-only overlaps, single-only regions, and the outside region."},{"text":"

Pair-only overlaps
Centre (all three sets)
Outside region
Single-only regions

","correct":false,"explanation":"Pair totals normally include the centre, so entering pair-only values before the centre risks overcounting. The correct progression is centre, pair-only, single-only, then outside."}]},{"id":"c1a5cf22-0124-4853-bf5e-c4d8b3c6f77f","cardType":"cloze","points":null,"front":"In a three-circle diagram, a statement like \"18 like rugby and football\" usually includes those who [ ].","back":"In a three-circle diagram, a statement like \"18 like rugby and football\" usually includes those who also like the third sport.","mcqOptions":[{"text":"also like the third sport","correct":true,"explanation":"An intersection of rugby and football includes everyone who likes both, including people who also like the third sport, unless the wording says \"only\"."},{"text":"do not like the third sport","correct":false,"explanation":"This incorrectly restricts the count to the rugby-and-football-only region. Without the word \"only\", the count also includes people who like the third sport."},{"text":"like exactly two sports","correct":false,"explanation":"\"Exactly two\" would exclude the central three-way intersection. A statement that people like rugby and football normally includes those who also like the third sport."},{"text":"like neither of those sports","correct":false,"explanation":"People who like neither rugby nor football cannot belong to their intersection. The statement describes everyone who likes both, including those who also like the third sport."}]},{"id":"5a8ff670-d7fe-eaa4-48b1-fb2eb16388eb","cardType":"basic","points":2,"front":"How do you calculate the probability of a region if elements are equally likely?","back":"$$$\\frac{\\text{Number of elements in the region}}{\\text{Total elements in the universal set}}$$","mcqOptions":[{"text":"$$$\\frac{\\text{Number of elements in the region}}{\\text{Total elements in the universal set}}$$","correct":true,"explanation":"For equally likely elements, probability is favourable outcomes divided by all possible outcomes. The region supplies the favourable count and the universal set supplies the total."},{"text":"$$$\\frac{\\text{Total elements in the universal set}}{\\text{Number of elements in the region}}$$","correct":false,"explanation":"This reverses the probability fraction and can produce a value greater than 1. Divide the number in the region by the total in the universal set."},{"text":"$$$\\frac{\\text{Number of elements outside the region}}{\\text{Total elements in the universal set}}$$","correct":false,"explanation":"Using the count outside the region calculates the complementary probability. The requested probability uses the number inside the region over the universal-set total."},{"text":"$$$\\frac{\\text{Number of elements in the region}}{\\text{Total elements outside the region}}$$","correct":false,"explanation":"The denominator must count every equally likely outcome, not merely those outside the region. Use the total number of elements in the universal set."}]},{"id":"c1a5be48-c5c4-1b0a-b871-1c8cc24757bd","cardType":"cloze","points":null,"front":"When shading the union $A \\cup B$, you must include the A-only region, the B-only region, and the [ ].","back":"When shading the union $A \\cup B$, you must include the A-only region, the B-only region, and the overlap.","mcqOptions":[{"text":"overlap","correct":true,"explanation":"A union contains every element in A or B or both. Therefore, its shading includes both single-set regions and the overlap between the circles."},{"text":"outside region","correct":false,"explanation":"The outside region contains elements in neither set, so it is excluded from $A \\cup B$. The union includes the A-only, B-only, and overlap regions."},{"text":"A-only region","correct":false,"explanation":"The A-only region has already been listed and does not replace the shared section. The remaining part needed to complete the union is the overlap."},{"text":"B-only region","correct":false,"explanation":"The B-only region has already been included, so naming it again omits elements belonging to both sets. The union must also contain the overlap."}]},{"id":"e1292c57-e13d-8744-7a28-c77ba8374ef9","cardType":"basic","points":2,"front":"Describe the region represented by $A' \\cup B$.","back":"Everything except the A-only region.","mcqOptions":[{"text":"Everything except the A-only region.","correct":true,"explanation":"$$A'$ includes everything outside A, while B restores the overlap because it belongs to B. Only the part in A but not B remains excluded."},{"text":"Everything except the B-only region.","correct":false,"explanation":"The B-only region belongs to both $A'$ and B, so it must be included. The only excluded area is the A-only region."},{"text":"Everything except the overlap region.","correct":false,"explanation":"Although the overlap is not in $A'$, it is in B and therefore belongs to the union. Only the A-only region is excluded."},{"text":"Everything except both circle regions.","correct":false,"explanation":"This describes the area outside both sets and wrongly excludes most of B. $A' \\cup B$ includes all regions except the A-only region."}]},{"id":"b9a1e4e4-8fcc-e06f-2225-c0971eca2bf8","cardType":"cloze","points":null,"front":"When placing listed elements into a diagram, every element from the universal set must be placed in [ ] region.","back":"When placing listed elements into a diagram, every element from the universal set must be placed in exactly one region.","mcqOptions":[{"text":"exactly one","correct":true,"explanation":"The regions of a Venn diagram are mutually exclusive areas, so each element in the universal set belongs in exactly one region, including any region outside the circles."},{"text":"one or more","correct":false,"explanation":"Allowing one or more regions would permit the same element to be recorded repeatedly. Each element must be placed in exactly one mutually exclusive region."},{"text":"exactly two","correct":false,"explanation":"Placing every element in two regions would double-count it and misrepresent its set membership. Each element belongs in exactly one region of the diagram."},{"text":"every","correct":false,"explanation":"An element should not appear throughout the diagram, because each region represents a particular combination of set memberships. It must be placed in exactly one region."}]},{"id":"ec04ce88-78df-cf27-0c8d-8615ad703a98","cardType":"cloze","points":null,"front":"To find the count for the \"neither\" region, [ ] the total number of elements inside all circles from the universal set total.","back":"To find the count for the \"neither\" region, subtract the total number of elements inside all circles from the universal set total.","mcqOptions":[{"text":"subtract","correct":true,"explanation":"Everything inside the circles is removed from the universal set total, leaving the number outside all circles. That remaining count represents neither set."},{"text":"estimate","correct":false,"explanation":"Estimating is unnecessary when the relevant totals are known and may produce an inaccurate count. Subtract the number inside all circles from the universal set total."},{"text":"separate","correct":false,"explanation":"Separating the two totals does not calculate the number in neither set. Subtract the complete count inside the circles from the universal set total."},{"text":"infer","correct":false,"explanation":"The neither count cannot be inferred without performing the required arithmetic. Subtract the number inside all circles from the universal set total to find it."}]},{"id":"576ecfeb-df8a-69de-426d-2c11c3587e43","cardType":"basic","points":1,"front":"What is the common mistake when shading the union $A \\cup B$?","back":"Shading only the overlap instead of both circles entirely.","mcqOptions":[{"text":"Shading only the overlap instead of both circles entirely.","correct":true,"explanation":"The union contains everything in A, in B, or in both sets. Shading only their overlap confuses the union with the intersection."},{"text":"Shading both circles entirely instead of only the overlap.","correct":false,"explanation":"Shading both circles entirely is the correct representation of the union, not the mistake. Shading only the overlap is the usual confusion with the intersection."},{"text":"Shading everything outside both circles instead of both circles entirely.","correct":false,"explanation":"The area outside both circles represents elements in neither set, not the union. The usual union error is shading only the overlap rather than both circles."},{"text":"Shading only circle A instead of both circles entirely.","correct":false,"explanation":"Shading only circle A omits elements that belong exclusively to B, but it is not the stated intersection confusion. The common error is shading only the overlap."}]},{"id":"a8ef2170-740c-9af9-9349-f7f7a62cb826","cardType":"basic","points":1,"front":"What must you do to every region to check if a Venn diagram is completed correctly?","back":"Sum every region (including the outside); it must equal the total number of items in the survey.","mcqOptions":[{"text":"Sum every region (including the outside); it must equal the total number of items in the survey.","correct":true,"explanation":"Every mutually exclusive region, including the area outside the circles, accounts for part of the universal set. Their sum must therefore equal the survey total."},{"text":"Sum every region (excluding the outside); it must equal the total number of items in the survey.","correct":false,"explanation":"Excluding the outside region omits respondents who belong to neither set. Sum every region, including the outside, and compare the result with the survey total."},{"text":"Multiply every region (including the outside); it must equal the total number of items in the survey.","correct":false,"explanation":"Region frequencies represent separate groups that must be added, not multiplied. The sum of all regions, including the outside, must equal the survey total."},{"text":"Sum only the circles (excluding the outside); it must equal the total number of items in the survey.","correct":false,"explanation":"Counting only the circles ignores items belonging to neither set, so it cannot verify the universal total. Every region, including the outside, must be summed."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Venn Diagrams","title":"Venn Diagrams Flashcards","topbar":{"backHref":"/gcse/maths/aqa/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"929a3f0c-f3a9-45f0-9f12-edba92b20630","walled":"$L39"}],"$L3a","$L3b"]